# A Hilbert space approach to singularities of functions

Jim Agler  Department of Mathematics, University of California at San Diego,
CA 92103, USA<EMAIL_ADDRESS>,  Zinaida A. Lykova  School of Mathematics,
Statistics and Physics, Newcastle University, Newcastle upon Tyne NE1 7RU,
U.K<EMAIL_ADDRESS>and  N. J. Young  School of Mathematics,
Statistics and Physics, Newcastle University, Newcastle upon Tyne NE1 7RU,
U.K. and School of Mathematics, Leeds University, Leeds LS2 9JT, U.K.
<EMAIL_ADDRESS>
(Date: 10th May, 2022)

###### Abstract.

We introduce the notion of a pseudomultiplier of a Hilbert space $\mathcal{H}$
of functions on a set $\Omega$. Roughly, a pseudomultiplier of $\mathcal{H}$
is a function which multiplies a finite-codimensional subspace of
$\mathcal{H}$ into $\mathcal{H}$, where we allow the possibility that a
pseudomultiplier is not defined on all of $\Omega$. A pseudomultiplier of
$\mathcal{H}$ has singularities, which comprise a subspace of $\mathcal{H}$,
and generalize the concept of singularities of an analytic function, even
though the elements of $\mathcal{H}$ need not enjoy any sort of analyticity.
We analyse the natures of these singularities, and obtain a broad
classification of them in function-theoretic terms.

###### Key words and phrases:

Reproducing kernel Hilbert space, Hilbert function space, multipliers

###### 2020 Mathematics Subject Classification:

46E22, 47B38, 46E20

Partially supported by National Science Foundation Grants DMS 1361720 and
1665260, a Newcastle URC Visiting Professorship, the Engineering and Physical
Sciences Research Council grant EP/N03242X/1 and London Mathematical Society
Grant 42013

###### Contents

  1. 1 Introduction
  2. 2 Overview of the paper
  3. 3 Examples of pseudomultipliers
  4. 4 Ambiguous points
  5. 5 Ambiguous vectors and the ambiguous space
  6. 6 Definable vectors and polar vectors
  7. 7 The singular space
  8. 8 Visible subspaces of a pseudomultiplier
  9. 9 Local subspaces over a subset of $\Omega$
  10. 10 Projectively complete Hilbert function spaces and local subspaces

##  1\. Introduction

Many different mathematical structures have been brought to bear on the study
of functions. In the last half-century notable theories which have yielded
insights into function-theoretic problems include those of locally convex
spaces, ordered vector spaces, commutative Banach algebras and the geometry of
Banach spaces. Perhaps the richest structure of all is that of Hilbert spaces;
when a function-theoretic problem can be formulated in terms of Hilbert space
geometry one can have high hopes of making progress on it. Accordingly, there
has been a substantial development of the theory of Hilbert function spaces
(also called reproducing kernel Hilbert spaces and functional Hilbert spaces)
[3], [14].

By a Hilbert function space on a set $\Omega$ we mean a Hilbert space
$\mathcal{H}$ whose elements are complex-valued functions on $\Omega$ such
that, for each $\lambda\in\Omega$, the linear functional $f\mapsto f(\lambda)$
is continuous on $\mathcal{H}$. It follows that, for each $\lambda\in\Omega$
there exists an element of $\mathcal{H}$, which we shall denote by
$k_{\lambda}$, such that $f(\lambda)=\left\langle f,k_{\lambda}\right\rangle$
for every $f\in\mathcal{H}$. We call $k_{\lambda}$ the kernel corresponding to
$\lambda$, and we call the function $k:\Omega\times\Omega\to\mathbb{C}$ given
by

$k(\mu,\lambda)=k_{\lambda}(\mu)\quad\mbox{ for all }\lambda,\mu\in\Omega$

the reproducing kernel of $\mathcal{H}$.

Even when we study problems which are not framed in Hilbert space terms we may
still be able to use Hilbert space methods. This seminal idea is the basis of
a substantial body of work stemming from papers by Sarason [13] and Adamyan,
Arov and Krein [1] which has had a strong influence not only in analysis but
also in control theory [9]. In these papers bounded analytic functions are
analysed and constructed in terms of their action as multipliers on the Hardy
space $H^{2}$.

Both the elements of $H^{2}$ and the multipliers of $H^{2}$ are functions
possessing a high degree of regularity, and one might expect that the same
assertion will be true of most Hilbert function spaces one is likely to
encounter, and that therefore operator-theoretic methods should be well
adapted to problems involving highly regular functions, but of less use for
the study of singularities of functions. However, the cited paper of Adamyan,
Arov and Krein and its forerunner by Akhiezer [2] shows that this is not
entirely true. One can construct meromorphic functions in the disc with a
prescribed number of poles by the spectral analysis of operators on $H^{2}$.
In this approach a function with singularities determines a multiplier not on
the whole of $H^{2}$ but on a closed subspace of finite codimension; such an
object is called a pseudomultiplier of the Hilbert space in question. The
power of operator theory in analysing functions with singularities is thus far
from vacuous, and we are led to wonder about its potential and its
limitations.

In [5] the domain of definition of a pseudomultiplier, its associated “Pick
kernel”, and the particular case of Hilbert spaces of analytic functions of
one variable were studied. The paper [6] is devoted to a key result of
Adamyan, Arov and Krein and the extent to which it can have analogues for
other function spaces. Pseudomultipliers also appear naturally in connection
with nearly invariant subspaces for backward shift operators on vector-valued
Hardy spaces (see [7]).

In this paper we will develop the theory of pseudomultipliers and local
subspaces. The main notions and results are presented in the overview of the
paper and illustrated by examples.

##  2\. Overview of the paper

Before giving a formal definition of a pseudomultiplier, let us consider some
archetypal examples. The Hardy space $H^{2}$ is the space of analytic
functions $f$ on the open unit disc $\mathbb{D}$ such that

$\sup_{0<r<1}\int_{0}^{2\pi}|f(re^{i\theta})|^{2}\ d\theta<\infty.$

When endowed with pointwise addition and scalar multiplication and the inner
product

$\left\langle f,g\right\rangle=\lim_{r\to
1-}\frac{1}{2\pi}\int_{0}^{2\pi}f(re^{i\theta})\overline{g(re^{i\theta})}\
d\theta$

$H^{2}$ is a Hilbert function space, whose reproducing kernel is the Szegő
kernel $k$, defined by

$k(\lambda,\mu)=\frac{1}{1-\bar{\mu}\lambda}\quad\mbox{ for all
}\lambda,\mu\in\mathbb{D}.$

An elementary introduction to $H^{2}$ can be found in [15, Chapter 13]. More
thorough and far-reaching treatments are in [10, 12].

2.1. Pseudomultipliers

######  Definition 2.1.

A multiplier of a Hilbert function space $\mathcal{H}$ on a set $\Omega$ is
defined to be a function $\varphi:\Omega\to\mathbb{C}$ such that $\varphi
f\in\mathcal{H}$ for every $f\in\mathcal{H}$. Here $\varphi f$ denotes the
pointwise product of $\varphi$ and $f$.

One can show directly from the definitions that every function in the space
$H^{\infty}$ of bounded analytic functions on $\mathbb{D}$ is a multiplier of
$H^{2}$.  On the other hand, if $\varphi$ is a multiplier of $H^{2}$, then
since the constant function $\mathbf{1}\in H^{2}$, we have
$\varphi=\varphi\mathbf{1}\in H^{2}$, and so $\varphi$ is analytic on
$\mathbb{D}$. Moreover, the operator $X_{\varphi}$ on $H^{2}$ defined, for all
$f\in H^{2}$, by

$(X_{\varphi}f)(\lambda)=\varphi(\lambda)f(\lambda),\quad\mbox{ for all
}\lambda\in\Omega$ (2.1)

is linear and is easily seen to have a closed graph, and hence $X_{\varphi}$
is a bounded linear operator on $H^{2}$. The calculation

$\displaystyle\left\langle X_{\varphi}^{*}k_{\lambda},f\right\rangle$
$\displaystyle=$ $\displaystyle\left\langle
k_{\lambda},X_{\varphi}f\right\rangle=\left\langle k_{\lambda},\varphi
f\right\rangle$ $\displaystyle=$ $\displaystyle\overline{(\varphi
f)(\lambda)}=\overline{\varphi(\lambda)}\left\langle
k_{\lambda},f\right\rangle\mbox{ for all }\lambda\in\mathbb{D}\;\text{ and all
}\;f\in H^{2}$

shows that
$X_{\varphi}^{*}k_{\lambda}=\overline{\varphi(\lambda)}k_{\lambda}$, so that
$\overline{\varphi(\lambda)}$ is an eigenvalue of $X_{\varphi}^{*}$, and
therefore $|\varphi(\lambda)|\leq\|X_{\varphi}\|$ for all
$\lambda\in\mathbb{D}$, which is to say that $\varphi$ is bounded on
$\mathbb{D}$. Thus the multipliers of $H^{2}$ are precisely the elements of
$H^{\infty}$.

######  Example 2.2.

A pseudomultiplier of $H^{2}$. Consider the function $\varphi(z)=1/z$, defined
on the set $\mathbb{D}\setminus\\{0\\}$. This function is clearly not a
multiplier of $H^{2}$, since it is unbounded on $\mathbb{D}$, nor even defined
on the whole of $\mathbb{D}$. On the other hand it is close to being a
multiplier in the sense that there is a finite-codimensional closed subspace
of $H^{2}$, namely $zH^{2}$, which is a closed subspace of codimension $1$ in
$H^{2}$ with the property that, for every $f\in zH^{2}$, $\varphi f$ is the
restriction to $\mathbb{D}\setminus\\{0\\}$ of a function in $H^{2}$.

######  Example 2.3.

Another pseudomultiplier of $H^{2}$. Choose distinct complex numbers $a,b$ and
define

$\psi(z)=\left\\{\begin{array}[]{lll}a&\mbox{ if }z\neq 0\\\ &\mbox{ }\\\
b&\mbox{ if }z=0.\end{array}\right.$

For any $f\in H^{2}$, it is easy to see that $\psi f\in H^{2}$ if and only if
$f(0)=0$, that is, if and only if $f$ is in the closed $1$-codimensional
subspace $zH^{2}$ of $H^{2}$. Hence $\psi$ is a pseudomultiplier of $H^{2}$.

In the light of these two examples we introduce the following notion.

######  Definition 2.4.

Let $\mathcal{H}$ be a Hilbert function space on a set $\Omega$. We say that a
function

$\varphi:D_{\varphi}\subset\Omega\to\mathbb{C}$

is a pseudomultiplier of $\mathcal{H}$ if

  1. (1)

$D_{\varphi}$ is a set of uniqueness for $\mathcal{H}$;

  2. (2)

the subspace $E_{\varphi}$ of $\mathcal{H}$, defined to be

$\left\\{h\in\mathcal{H}:{\rm there\ exists}\ g\in\mathcal{H}\ {\rm such\
that}\ g(\lambda)=\varphi(\lambda)h(\lambda)\ {\rm for\ all}\ \lambda\in
D_{\varphi}\right\\}$

is closed in $\mathcal{H}$, and

  3. (3)

$E_{\varphi}$ has finite codimension in $\mathcal{H}$.

If $\mathcal{H}$ is a Hilbert function space on a set $\Omega$ then we say
that a subset $D$ of $\Omega$ is a set of uniqueness for $\mathcal{H}$ if, for
any functions $f,g\in\mathcal{H}$, if $f(\lambda)=g(\lambda)$ for all
$\lambda\in D$ then $f=g$.

When conditions (1) to (3) hold we define the operator
$X_{\varphi}:E_{\varphi}\to\mathcal{H}$ by $X_{\varphi}h=g$ where $g$ is the
element of $\mathcal{H}$ (necessarily unique, by condition (1)) such that
$g(\lambda)=\varphi(\lambda)h(\lambda)$ for all $\lambda\in D_{\varphi}$. The
space $E_{\varphi}$ will be called the regular space of $\varphi$, while the
space $E_{\varphi}^{\perp}$ will be called the singular space of $\varphi$ and
will be denoted by $\mathcal{S}_{\varphi}$.

For the pseudomultiplier $\varphi(z)=1/z$ on $H^{2}$ of Example 2.2,
$D_{\varphi}=\mathbb{D}\setminus\\{0\\}$ and $E_{\varphi}$ is the closed
subspace $zH^{2}$ of $H^{2}$, which has codimension $1$ in $H^{2}$. Thus
$\mathcal{S}_{\varphi}=H^{2}\ominus zH^{2}$, the subspace of constant
functions in $H^{2}$. The operator $X_{\varphi}:zH^{2}\to H^{2}$ is given by
$X_{\varphi}zf=f$ for all $f\in H^{2}$. For the pseudomultiplier $\psi$ of
$H^{2}$ of Example 2.3, $D_{\psi}=\mathbb{D}$, $E_{\psi}=zH^{2}$,
$\mathcal{S}_{\psi}$ is the subspace of constant functions in $H^{2}$ and
$X_{\psi}$ is the scalar operator $X_{\psi}f=af$ on $zH^{2}$.

The Hilbert function spaces on which pseudomultipliers act need not be spaces
of analytic functions.

######  Example 2.5.

A pseudomultiplier on a Sobolev space. Let $\mathcal{H}$ be the space
$W^{1,2}[0,1]$ of absolutely continuous functions on $[0,1]$ whose weak
derivatives are in $L^{2}$ and with inner product given by

$\left\langle
f,g\right\rangle=\int^{1}_{0}f(t)\overline{g(t)}dt+\int^{1}_{0}f^{\prime}(t)\overline{g^{\prime}(t)}dt.$

It is well known that $W^{1,2}[0,1]$ is a reproducing kernel Hilbert space
with reproducing kernel

$\displaystyle k(\lambda,\mu)=\left\\{\begin{array}[]{ll}{\rm
cosech}1\cosh(1-\mu)\cosh\lambda&\mbox{if $0\leq\lambda\leq\mu\leq 1,$}\\\
{\rm cosech}1\cosh(1-\lambda)\cosh\mu&\mbox{if $0\leq\mu\leq\lambda\leq
1$.}\end{array}\right.$ (2.4)

Let us check that $\chi(t)=\sqrt{t}$ is a pseudomultiplier of $\mathcal{H}$
with $E_{\chi}=k^{\perp}_{0}$ and $\mathcal{S}_{\chi}=\mathbb{C}k_{0}$.
Consider a function $f\in k_{0}^{\perp}$, so that $f\in W^{1,2}[0,1]$ and
$f(0)=0$. Let us prove that the function $\chi f$ is absolutely continuous and
$(\chi f)^{\prime}$ is in $L^{2}(0,1)$. Note that

$(\chi f)^{\prime}(t)=\frac{1}{2\sqrt{t}}f(t)+\sqrt{t}f^{\prime}(t).$

To prove that $(\chi f)^{\prime}$ is in $L^{2}(0,1)$, it is enough to show
that $\frac{1}{\sqrt{t}}f(t)$ is in $L^{2}(0,1)$. Since $f(0)=0$, for
$x\in(0,1)$,

$\displaystyle x^{-\tfrac{1}{2}}f(x)$ $\displaystyle=$ $\displaystyle
x^{-\tfrac{1}{2}}\int_{0}^{x}f^{\prime}(t)\,dt$ (2.5) $\displaystyle=$
$\displaystyle\sqrt{x}\left(x^{-1}\int_{0}^{x}f^{\prime}(t)\,dt\right).$

By assumption $f^{\prime}$ is in $L^{2}(0,1)$, and so, by Hardy’s inequality,
$\left(x^{-1}\int_{0}^{x}f^{\prime}(t)\,dt\right)$ is in $L^{2}(0,1)$. Thus
$\frac{1}{\sqrt{t}}f(t)$ is in $L^{2}(0,1)$. Hence $(\chi f)^{\prime}$ is in
$L^{2}(0,1)$. Therefore $(\chi f)^{\prime}\in L^{1}(0,1)$, and so

$(\chi f)(x)=\int_{0}^{x}(\chi f)^{\prime}(t)\ dt$

is absolutely continuous on $[0,1]$. Therefore, for $f\in k_{0}^{\perp}$,
$\chi f\in W^{1,2}$. Conversely, if $f(0)\neq 0$ then $(\chi f)^{\prime}\notin
L^{2}[0,1]$ and so $\chi f\notin W^{1,2}[0,1]$. Thus, $\chi$ is a
pseudomultiplier of $W^{1,2}$, $E_{\chi}=k_{0}^{\perp}$ and
$\mathcal{S}_{\chi}=\mathbb{C}k_{0}$. $\square$

Nor need it be true that the kernel vectors
$\\{k_{\lambda}:\lambda\in\Omega\\}$ are linearly independent in
$\mathcal{H}$.

######  Example 2.6.

Linearly dependent kernels. Let $a,b$ be distinct points in $\mathbb{D}$ and
let $\mathcal{H}$ comprise the functions $\\{f\in H^{2}:f(a)=f(b)\\}$, with
the inner product inherited from $H^{2}$. Thus, if $k$ is the reproducing
kernel of $\mathcal{H}$, we have $k_{a}=k_{b}$. Consider an element
$\upsilon\in H^{\infty}$ such that $\upsilon(a)\neq\upsilon(b)$. It is clear
that, for any $f\in\mathcal{H}$, $\upsilon f\in\mathcal{H}$ if and only if
$f(a)=f(b)=0$. Thus $E_{\upsilon}=k_{a}^{\perp}=k_{b}^{\perp}$, a closed
subspace of $\mathcal{H}$ of codimension $1$, and $\upsilon$ is a
pseudomultiplier of $\mathcal{H}$.

2.2. The singular space

Informally speaking, the singular space $\mathcal{S}_{\varphi}$ is what stops
the pseudomultiplier $\varphi$ from being a multiplier. Pseudomultipliers that
are not multipliers always have singularities in the sense that
$\mathcal{S}_{\varphi}$ is non-zero. The reader will have noticed that in the
three examples 2.2, 2.3 and 2.5 the pseudomultipliers $\varphi,\psi$, and
$\chi$ all fail to be multipliers of $\mathcal{H}$, in each case on account of
a lack of regularity at $0$; $\varphi$ is unbounded in a neighborhood of $0$,
$\psi$ is discontinuous at $0$ and $\chi$ is not locally in $L^{2}$ at $0$.
These examples motivated us to call $\mathcal{S}_{\varphi}$ the singular space
of $\varphi$.

In this paper we develop a structure theory for the singular spaces of
pseudomultipliers. The principal result is that $\mathcal{S}_{\varphi}$ can be
orthogonally decomposed into the direct sum of the polar space of $\varphi$,
which we denote by $\mathcal{P}_{\varphi}$, and the set of ambiguous vectors
of $\varphi$, which we denote by $\mathcal{A}_{\varphi}$. The definitions of
$\mathcal{P}_{\varphi}$ and $\mathcal{A}_{\varphi}$, which we shall give
shortly, depend on the notion of $\varphi$ seeing a vector.

2.3 Seeing a vector

Let $\mathcal{H}$ be a Hilbert function space on $\Omega$ and fix
$v\in\mathcal{H}$. We shall formalise the idea of enlarging the set $\Omega$
by adjoining an extra point $p\notin\Omega$ in such a way that the vector $v$
is the reproducing kernel associated with $p$. Let
$\tilde{\Omega}=\Omega\cup\\{p\\}$, and for $f\in\mathcal{H}$ define
$f_{v}:\tilde{\Omega}\to\mathbb{C}$ by the formula

$f_{v}(\lambda)=\left\\{\begin{array}[]{ll}f(\lambda)&{\rm
if}\qquad\lambda\in\Omega\\\ \\\ \left\langle f,v\right\rangle&{\rm if}\
\lambda=p\ .\end{array}\right.$

Let $\mathcal{H}_{v}=\\{f_{v}:f\in\mathcal{H}\\}$. Since $f=0$ in
$\mathcal{H}$ implies that $f_{v}=0$ in $\mathcal{H}_{v}$, we see that the
formula

$\left\langle f_{v},g_{v}\right\rangle_{\mathcal{H}_{v}}\ =\ \left\langle
f,g\right\rangle_{\mathcal{H}}$

defines an inner product on $\mathcal{H}_{v}$. Furthermore, with this inner
product, $\mathcal{H}_{v}$ is a Hilbert function space on $\tilde{\Omega}$,
and the restriction map

$\mathcal{H}_{v}\ni f_{v}\mapsto f_{v}|\Omega=f\in\mathcal{H}$

is a Hilbert space isomorphism.

Now let $\mathcal{H}$ be a Hilbert function space, fix $v\in\mathcal{H}$, and
let $\varphi$ be a pseudomultiplier of $\mathcal{H}$. Since $D_{\varphi}$ is a
set of uniqueness for $\mathcal{H}$, $D_{\varphi}$ is also a set of uniqueness
for $\mathcal{H}_{v}$ and thus $\varphi$ is also a pseudomultiplier of
$\mathcal{H}_{v}$.  Its domain when so regarded is still just $D_{\varphi}$
and its regular space is unchanged. One can wonder, however, whether $\varphi$
can be extended to a pseudomultiplier of $\mathcal{H}_{v}$ with domain
containing $D_{\varphi}\cup\\{p\\}$.

######  Definition 2.7.

Let $\mathcal{H}$ be a Hilbert function space on $\Omega$, let
$v\in\mathcal{H}$, let $c\in\mathbb{C}$, and let $\varphi$ be a
pseudomultiplier of $\mathcal{H}$. We say that $\varphi$ sees $v$ with value
$c$ if there exists a pseudomultiplier $\psi$ of $\mathcal{H}_{v}$ such that
$\psi$ extends $\varphi$ (that is, $D_{\varphi}\subseteq D_{\psi}$ and
$\varphi=\psi|D_{\varphi}$), $p\in D_{\psi},E_{\varphi}=E_{\psi}|\Omega$, and
$\psi(p)=c$. We say $\varphi$ sees $v$ if there exists $c\in\mathbb{C}$ such
that $\varphi$ sees $v$ with value $c$; in this case we also say that $v$ is
visible to $\varphi$.

######  Example 2.8.

Let $\mathcal{H}$ be a Hilbert function space on $\Omega$ and let $\varphi$ be
a pseudomultiplier of $\mathcal{H}$. For any $c\in\mathbb{C},\varphi$ sees the
zero vector in $\mathcal{H}$ with value $c$. To prove this fact, in Definition
2.7 choose $v=0$ and $\psi$ to be given by $\psi(\lambda)=\varphi(\lambda)$
for $\lambda\in\Omega$ and $\psi(p)=c$. One finds that $f_{v}\in E_{\psi}$ if
and only if $f\in E_{\varphi}$ and that $\psi$ is a pseudomultiplier of
$\mathcal{H}_{v}$. Thus $\varphi$ sees $0$ with value $c$.

######  Example 2.9.

Let $\mathcal{H}$ be a Hilbert function space on $\Omega$ and let $\varphi$ be
a pseudomultiplier of $\mathcal{H}$. If $\lambda\in D_{\varphi}$, then
$\varphi$ sees $k_{\lambda}$ with value $\varphi(\lambda)$.

To prove this statement, let $\tilde{\Omega}=\Omega\cup\\{p\\}$, where
$p\notin\Omega$, and let $D_{\psi}=D_{\varphi}\cup\\{p\\}$. Define a function
$\psi:D_{\psi}\to\mathbb{C}$ by $\psi(z)=\varphi(z)$ if $z\in D_{\varphi}$ and
$\psi(p)=\varphi(\lambda)$. Let $v=k_{\lambda}$, so that $\mathcal{H}_{v}$ is
a Hilbert space of functions on $\tilde{\Omega}$. We claim that $\psi$ is a
pseudomultiplier of $\mathcal{H}_{v}$ and that $E_{\varphi}=E_{\psi}|\Omega$.
Indeed, consider any function $f\in\mathcal{H}$. We have

$\displaystyle f_{v}\in E_{\psi}$ $\displaystyle\Leftrightarrow$
$\displaystyle\psi f_{v}\in\mathcal{H}_{v}|_{D_{\psi}}$
$\displaystyle\Leftrightarrow$ $\displaystyle\text{there exists
}\;g\in\mathcal{H}\;\;\text{such that}\;(\psi f)_{v}=g_{v}|_{D_{\psi}}$

Now, for any $g\in\mathcal{H}$, $(\psi f)_{v}=g_{v}|_{D_{\psi}}$ if and only
if $\varphi(z)f(z)=g(z)$ for $z\in D_{\varphi}$ and
$\psi(p)f_{v}(p)=\left\langle g,v\right\rangle$, which is to say that $\varphi
f=g$ on $D_{\varphi}$ and $\varphi(\lambda)\left\langle
f,v\right\rangle=\left\langle g,v\right\rangle$. Thus $f_{v}\in E_{\psi}$ if
and only if $\varphi f$ extends to an element $g\in\mathcal{H}$ and
$\varphi(\lambda)f(\lambda)=g(\lambda)$. Thus $f_{v}\in E_{\psi}$ if and only
if $f\in E_{\varphi}$. Hence, by Definition 2.7, $\varphi$ sees $k_{\lambda}$
with value $\varphi(\lambda)$, as claimed. $\square$

######  Example 2.10.

What vectors does $1/z$ see in $H^{2}$? Consider the pseudomultiplier
$\varphi$ of Example 2.2: $\varphi(z)=1/z$ on
$(\mathcal{H},\Omega)=(H^{2},\mathbb{D})$. Here,
$D_{\varphi}=\mathbb{D}\setminus\\{0\\}$ and $E_{\varphi}=zH^{2}$. We claim
that the vectors in $\mathcal{H}=H^{2}$ visible to $\varphi$ are precisely the
scalar multiples of the kernels $k_{\lambda}$, for
$\lambda\in\mathbb{D}\setminus\\{0\\}.$

To prove this claim, we first note that, by Example 2.9, $\varphi$ does see
$k_{\lambda}$ with value $\varphi(\lambda)=1/\lambda$ for every $\lambda\in
D_{\varphi}$. Conversely, suppose $\varphi$ sees a vector $v\neq 0$ with value
$c$. By definition, there exists a pseudomultiplier $\psi$ of
$\mathcal{H}_{v}$ such that $\psi$ extends $\varphi$, $p\in
D_{\psi},E_{\varphi}=E_{\psi}|\mathbb{D}$, and $\psi(p)=c$. Consider any
function $h\in H^{2}$. The function $f$ defined by $f(z)=zh(z)$ is in
$E_{\varphi}=zH^{2}$. Hence there exists $\tilde{f}\in E_{\psi}$ such that
$\tilde{f}|\mathbb{D}\setminus\\{0\\}=f$. We have
$\psi\tilde{f}\in\mathcal{H}_{v}$, and therefore

$(\psi\tilde{f})(p)=\left\langle(\psi\tilde{f})|\mathbb{D}\setminus\\{0\\},v\right\rangle=\left\langle\tfrac{1}{z}f,v\right\rangle=\left\langle\tfrac{1}{z}zh,v\right\rangle=\left\langle
h,v\right\rangle.$

Now $(\psi\tilde{f})(p)=\psi(p)\tilde{f}(p)=c\left\langle
f,v\right\rangle=c\left\langle zh,v\right\rangle$. Therefore, for every $h\in
H^{2}$, $c\left\langle zh,v\right\rangle=\left\langle h,v\right\rangle$, and
so

$\left\langle h,v\right\rangle=\left\langle
h,\bar{c}\bar{z}v\right\rangle=\left\langle
h,\bar{c}P_{+}\bar{z}v\right\rangle=\left\langle
h,\bar{c}S^{*}v\right\rangle,$

where $P_{+}:L^{2}(\mathbb{T})\to H^{2}$ is the orthogonal projection operator
and $S^{*}$ is the backward shift operator on $H^{2}$, given by
$S^{*}g=P_{+}(\bar{z}g)$. Hence $\bar{c}S^{*}v=v$. Since, by assumption,
$v\neq 0$, $v$ is an eigenvector of $S^{*}$ corresponding to the eigenvalue
$1/\bar{c}$. It is a simple calculation to show (see for example [4, Lemma
1.67]) that the eigenvectors of $S^{*}$ are the non-zero scalar multiples of
the Szegő kernels $k_{\lambda}$ for $\lambda\in\mathbb{D}$, that is,
$S^{*}k_{\lambda}=\bar{\lambda}k_{\lambda}$. Hence
$\bar{c}\bar{\lambda}k_{\lambda}=k_{\lambda}$, and so $c\lambda=1$, and we
conclude that $\lambda\in\mathbb{D}\setminus\\{0\\}$, and that $v$ is a scalar
multiple of the kernel $k_{\lambda}$, for some
$\lambda\in\mathbb{D}\setminus\\{0\\}$, as claimed. $\square$

2.4. Ambiguous points and the ambiguous space

In order to define the ambiguous space $\mathcal{A}_{\varphi}$ of a
pseudomultiplier $\varphi$ we first consider the points in $\Omega$ at which
the value of $\varphi$ is not determined by the operator $X_{\varphi}$. We
call such points ambiguous points of $\varphi$. For $c\in\mathbb{C}$ and
$\alpha\in D_{\varphi}$, define the function $\varphi_{c,\alpha}$ on
$D_{\varphi}$ by

$\varphi_{c,\alpha}(\lambda)=\left\\{\begin{array}[]{cl}\varphi(\lambda)&\ \
{\rm if}\quad\lambda\in D_{\varphi}\setminus\\{\alpha\\}\\\ \\\ c&\ \ {\rm
if}\quad\lambda=\alpha.\end{array}\right.$ (2.6)

Typically, modification of $\varphi$ in this manner will affect the operator
$X_{\varphi}:E_{\varphi}\to\mathcal{H}$. This modification could happen in two
possible ways: $E_{\varphi}$ could be changed, or $E_{\varphi}$ could remain
unaltered but the action of $X_{\varphi}$ could be changed.

######  Definition 2.11.

Let $\varphi$ be a pseudomultiplier and let $\alpha\in D_{\varphi}$. We say
$\alpha$ is an ambiguous point for  $\varphi$, or an ambiguity of $\varphi$,
if there exists $c\in\mathbb{C}$ with the properties that
$c\not=\varphi(\alpha),\;E_{\varphi}=E_{\varphi_{c,\alpha}}$, and
$X_{\varphi_{c,\alpha}}=X_{\varphi}$. We say $\alpha$ is an unambiguous point
for $\varphi$ if $\alpha$ is not an ambiguous point for $\varphi$.

The pseudomultiplier $\psi$ of $H^{2}$ given in Example 2.3 has a
discontinuity at $0$ which is an ambiguous point for $\psi$. Indeed, for any
choice of $c\in\mathbb{C}\setminus\\{a,b\\}$, the one-point modification
$\psi_{c,0}$ of $\psi$ is still an instance of Example 2.3, so that
$E_{\psi_{c,0}}=k_{0}^{\perp}=E_{\psi}$. Moreover, for any $f\in
k_{0}^{\perp}$, $X_{\psi}f=af=X_{\psi_{c,0}}f$. Another example of an
ambiguous point is furnished by the pseudomultiplier $\chi$ in Example 2.5. It
has an ambiguous point at $0$, as we show in Example 4.2 below.

In Section 4, Theorem 4.14, we shall prove the following connection between
visibility of vectors and ambiguous points.

######  Theorem 2.12.

Let $\varphi$ be a pseudomultiplier of a Hilbert function space $\mathcal{H}$
on a set $\Omega$ and let $\alpha\in\Omega$. Then $\alpha$ is an ambiguous
point for $\varphi$ if and only if $\varphi$ sees $k_{\alpha}$ with arbitrary
value.

Theorem 2.12 leads naturally to a more general notion, an ambiguous vector for
a pseudomultiplier.

######  Definition 2.13.

Let $\varphi$ be a pseudomultiplier of $\mathcal{H}$ and let
$v\in\mathcal{H}$. We say that $v$ is an ambiguous vector for  $\varphi$ if
$\varphi$ sees $v$ with arbitrary value, and we define $\mathcal{A}_{\varphi}$
to be the set of ambiguous vectors for $\varphi$.

It follows from Theorem 2.12 that if $\alpha$ is a ambiguous point for
$\varphi$ then $k_{\alpha}$ is an ambiguous vector for $\varphi$. Observe also
that Example 2.8 immediately implies that, for any Hilbert function space
$\mathcal{H}$, the zero vector of $\mathcal{H}$ is an ambiguous vector of
every pseudomultiplier of $\mathcal{H}$.

######  Example 2.14.

$\varphi(z)=1/z$ on $H^{2}$ has no non-zero ambiguities. By Example 2.10, the
pseudomultiplier $\varphi(z)=1/z$ of $H^{2}$ sees only the scalar multiples of
kernels $k_{\lambda}$ for some $\lambda\in\mathbb{D}\setminus\\{0\\}$.
Moreover, for such $\lambda$, $\varphi$ sees $k_{\lambda}$ only with value
$\varphi(\lambda)$. Thus $\varphi$ does not see any non-zero vector with
arbitrary value, which is to say that $\varphi$ has no ambiguous vectors other
than $0$.

######  Example 2.15.

Consider the pseudomultiplier $\chi(t)=\sqrt{t}$ on $W^{1,2}[0,1]$ of Example
2.5 We will show in Example 5.5, that $\mathcal{A}_{\chi}=\mathbb{C}k_{0}$.

2.5. Definable vectors and the polar space

In addition to singularities that arise from ambiguous vectors,
pseudomultipliers can possess another type of singular vector, which we call a
polar vector. We say that a vector $d\in\mathcal{H}$ is definable for
$\varphi$ if $d\not=0,\varphi$ sees $d$, and $d\perp\mathcal{A}_{\varphi}$. We
denote by $\mathbf{D}_{\varphi}$ the set of vectors that are definable for
$\varphi$. For any definable vector $v$ for a pseudomultiplier $\varphi$,
there is a unique $c\in\mathbb{C}$ such that $\varphi$ sees $v$ with value
$c$. We denote this number $c$ by $\varphi(v)$. Thus we can regard $\varphi$
as a function on $\mathbf{D}_{\varphi}$.

######  Example 2.16.

The definable vectors for $1/z$ on $H^{2}$. By Example 2.14, for the
pseudomultiplier $\varphi(z)=1/z$ of $H^{2}$, $\mathcal{A}_{\varphi}=\\{0\\}$,
and so the definable vectors for $\varphi$ coincide with the non-zero vectors
visible to $\varphi$. Hence, by Example 2.10, $\mathbf{D}_{\varphi}$ comprises
the non-zero scalar multiples of the kernels $k_{\lambda}$, for
$\lambda\in\mathbb{D}\setminus\\{0\\}$.

######  Definition 2.17.

Let $\varphi$ be a pseudomultiplier and let $p\in\mathcal{H}$. We say $p$ is a
polar vector for $\varphi$ if $p\not=0$ and there exists a sequence of
definable vectors $\\{d_{n}\\}\subseteq\mathbf{D}_{\varphi}$ such that
$d_{n}\to p$ (with respect to $\|\cdot\|_{\mathcal{H}}$) and
$\varphi(d_{n})\to\infty$ as $n\to\infty$. We define $\mathcal{P}_{\varphi}$,
the polar space of $\varphi$, to be the set of vectors $v\in\mathcal{H}$ such
that either $v$ is a polar vector of $\varphi$ or $v=0$.

######  Remark 2.18.

In [5, Section 1] we introduced a notion more directly modelled on the
familiar notion of a pole from Complex Analysis. If $\varphi$ is a
pseudomultiplier of a Hilbert function space $\mathcal{H}$ on a set $\Omega$
we said in [5] that a point $\alpha\in\Omega$ is a pseudopole of $\varphi$ if
there exist functions $g\in\mathcal{H}$ and $h\in E_{\varphi}$ such that
$h(\alpha)=0$, $g$ extends $\varphi h$ and $g(\alpha)\neq 0$.

From the perspective of this paper, pseudopoles are superseded by the more
general notion of polar vectors. We shall prove in Proposition 6.8 that if
$\alpha$ is a pseudopole of a pseudomultiplier $\varphi$ and $\alpha$ is in
the closure of $D_{\varphi}$ in $\Omega$ then
$P_{\mathcal{A}_{\varphi}^{\perp}}k_{\alpha}$ is a polar vector of $\varphi$.
$\square$

######  Example 2.19.

The polar space of $1/z$. In Example 2.16 we proved that the set of definable
vectors $\mathbf{D}_{\varphi}$ for $\varphi(z)=1/z$ on $H^{2}$ comprises the
scalar multiples of the kernels $k_{\lambda}$, for
$\lambda\in\mathbb{D}\setminus\\{0\\}$. Recall that the singular space
$\mathcal{S}_{\varphi}=E_{\varphi}^{\perp}=H^{2}\ominus zH^{2}$. Let us show
that

$\mathcal{P}_{\varphi}=\mathcal{S}_{\varphi}=E_{\varphi}^{\perp}=H^{2}\ominus
zH^{2}=\\{\text{constant functions on}\;\mathbb{D}\\}.$

It is enough to show that $1=k_{0}$ is in $\mathcal{P}_{\varphi}$. Choose
$\lambda_{n}\in\mathbb{D}$, $n\geq 1$, such that $\lambda_{n}\to 0$. By
Example 2.9, $\varphi$ sees $k_{\lambda_{n}}$ with value
$\frac{1}{\lambda_{n}}$. Since $k_{\lambda_{n}}\to k_{\mu}$ in $H^{2}$ if and
only if $\lambda_{n}\to\mu$ in the usual topology of $\mathbb{D}$,
$\\{k_{\lambda_{n}}\\}$ is a sequence of definable vectors such that
$k_{\lambda_{n}}\to k_{0}$ in $H^{2}$ and
$\varphi(k_{\lambda_{n}})=\varphi(\lambda_{n})=\frac{1}{\lambda_{n}}\to\infty$.
This implies that $k_{0}\in\mathcal{P}_{\varphi}$. $\square$

In Section 6, Theorem 6.10, we shall prove the following fact.

######  Theorem 2.20.

Let $\varphi$ be a pseudomultiplier of $\mathcal{H}$. Then $\varphi$ is
locally bounded on $\mathbf{D}_{\varphi}^{-}$ if and only if $\varphi$ has no
polar vectors.

In Section 6, Theorem 6.11, we shall also establish an alternative description
of the polar space in terms of definable vectors.

######  Theorem 2.21.

If $\mathcal{H}$ is infinite-dimensional and $\varphi$ is a pseudomultiplier
of $\mathcal{H}$ then
$\mathcal{P}_{\varphi}=\overline{\mathbf{D}_{\varphi}}\setminus\mathbf{D}_{\varphi}$.

2.6. Decomposition of the singular space

In Theorem 7.2 we shall prove that every singular vector of a pseudomultiplier
can be presented in a unique way as the sum of a polar vector and an ambiguous
vector.

######  Theorem 2.22.

Let $\mathcal{H}$ be a Hilbert function space on a set $\Omega$. Let $\varphi$
be a pseudomultiplier of $\mathcal{H}$. $\mathcal{P}_{\varphi}$ is a closed
subspace of $\mathcal{H}$ and

$\mathcal{S}_{\varphi}=\mathcal{P}_{\varphi}\oplus\mathcal{A}_{\varphi}.$

This theorem contains the very interesting fact that the polar space
$\mathcal{P}_{\varphi}$, which is the set of polar vectors of a
pseudomultiplier $\varphi$, together with the zero vector, is a linear
subspace of $\mathcal{H}$.

######  Example 2.23.

The pseudomultiplier $\varphi(z)=\frac{1}{z(z-{\tfrac{1}{2}})}$ on $H^{2}$ has
no ambiguities $\mathcal{A}_{\varphi}=\\{0\\}$, and the polar space of
$\varphi$

$\mathcal{P}_{\varphi}=\operatorname{span}\\{k_{0},k_{{\tfrac{1}{2}}}\\}=H^{2}\ominus
z(z-{\tfrac{1}{2}})H^{2}=E_{\varphi}^{\perp}.$

See Example 7.3 for proofs.

######  Example 2.24.

If $\chi(t)=\sqrt{t}$ on $W^{1,2}[0,1]$ is the pseudomultiplier of Example
2.5, then, as Example 2.15 implies, the ambiguous space
$\mathcal{A}_{\chi}=\mathbb{C}k_{0}=\mathcal{S}_{\chi}$, and so we deduce from
Theorem 7.2 that $\mathcal{P}_{\chi}=\\{0\\}$. Thus $\chi$ has no polar
vectors. This example illustrates Theorem 2.20.

2.7. Locality

In Section 8 we shall introduce the notion of a visible subspace for a
pseudomultiplier. We generalise the notion of $\varphi$ seeing a vector to
$\varphi$ seeing a closed subspace of $\mathcal{H}$. We say that $\varphi$
sees a closed subspace $\mathcal{V}$ if there is a bounded linear operator
$C:\mathcal{V}\to\mathcal{V}$ such that, for all $f\in E_{\varphi}$,

$P_{\mathcal{V}}X_{\varphi}f=CP_{\mathcal{V}}f.$ (2.7)

######  Example 2.25.

Let $\varphi\in H^{\infty}$, so that $\varphi$ is a multiplier of $H^{2}$. The
only vectors in $H^{2}$ seen by $\varphi$ are the eigenvectors of
$X^{*}_{\varphi}$. In particular, if $\varphi$ is such that
$\varphi^{\prime}(0)\neq 0$, then $\varphi$ does not see the vector
$v(\lambda)=\lambda,\lambda\in\mathbb{D}$. However, $\varphi$ does see the
space $\mathcal{V}\stackrel{{\scriptstyle\rm
def}}{{=}}\operatorname{span}\\{1,v\\}$: for $f\in
H^{2},P_{\mathcal{V}}X_{\varphi}f$ depends only the first two Taylor
coefficients of $f$, that is, on $P_{\mathcal{V}}f$. Thus $\varphi$ sees
$\mathcal{V}$, and there is a unique $C$ such that equation (8.1) holds for
all $f\in E_{\varphi}$. $\square$

We observe that in the case when $\varphi$ is a multiplier of $\mathcal{H}$,
$\varphi$ sees a closed subspace $\mathcal{V}$ of $\mathcal{H}$ if and only if
$\mathcal{V}$ is an invariant subspace for $X^{*}_{\varphi}$. In Proposition
8.8 we show that a closed subspace $\mathcal{V}$ is visible to a
psudomultiplier $\varphi$ if and only if
$X_{\varphi}^{*}\mathcal{V}\subseteq\mathcal{V}+E_{\varphi}^{\perp}$.

In Section 9 we introduce the idea of a local subspace of a Hilbert function
space over a subset $D$ of $\Omega$. Consider the subspace
$\mathcal{M}=\operatorname{span}\\{1,z\\}$ of $H^{2}$. The elements of
$\mathcal{M}^{\perp}$ are the functions which vanish together with their first
derivatives at $0$, and we accordingly regard $\mathcal{M}$ as being local at
$0$. Notice that $\mathcal{M}$ is not spanned by kernel functions – rather, it
is a limit of two-dimensional spaces spanned by kernels. For $f\in H^{2}$ we
have

$f^{\prime}(0)=\lim\limits_{\alpha\to
0}\frac{f(\alpha)-f(0)}{\alpha}=\lim\limits_{\alpha\to 0}\left\langle
f,\frac{k_{\alpha}-k_{0}}{\overline{\alpha}}\right\rangle.$

In a sense to be made precise, $\mathcal{M}$ is the limit as $\alpha\to 0$ of
the space $\operatorname{span}\\{1,\overline{\alpha}^{-1}(k_{\alpha}-1)\\}$
(which equals $\operatorname{span}\\{1,k_{\alpha}\\}$). Likewise
$\operatorname{span}\\{1,z,z^{2}\\}$ is the limit as $\alpha,\beta\to 0$ of
$\operatorname{span}\\{1,k_{\alpha},k_{\beta}\\}$. This example provides the
model for the notion of a local subspace of a general Hilbert function space.

Let $D\subset\Omega$. A closed subspace $\mathcal{M}$ of $\mathcal{H}$ is
local over $D$ if $\mathcal{M}$ is a cluster point of a collection of spans of
uniformly boundedly many kernels of points in $D$ (see Definition 9.8). We say
that $\alpha\in\Omega$ is a support point  of $\mathcal{M}$ if there exist a
positive integer $n$, a sequence $(\alpha_{j})$ in $\Omega$, a sequence
$(c_{j})$ in $\mathbb{C}$ and a sequence $(E_{j})$ of subsets of $\Omega$ such
that $\\#E_{j}=n,\alpha_{j}\in
E_{j},\operatorname{span}\\{k_{\lambda}:\lambda\in E_{j}\\}\to\mathcal{M}$ and
$c_{j}k_{\alpha_{j}}\to k_{\alpha}$. The support of $\mathcal{M}$ is the set
of support points of $\mathcal{M}$. It will be denoted by
$\operatorname{supp}\mathcal{M}$. We say that $\mathcal{M}$ is punctual if
$\operatorname{supp}\mathcal{M}$ is a singleton set.

The main result of Section 9 is Theorem 9.11, which asserts that all local
spaces are seen by all pseudomultipliers, as follows.

######  Theorem 2.26.

Let $\varphi$ be a pseudomultiplier of $\mathcal{H}$. If $\mathcal{M}$ is a
subspace of $\mathcal{H}$ which is local over $D_{\varphi}$, then $\varphi$
sees $\mathcal{M}$.

In light of Theorem 2.26 it appears that the structure of local subspaces is
an important component of the theory of pseudomultipliers. Therefore, in the
final section of the paper we study such local subspaces. In Theorem 10.13, we
establish the structure of local subspaces $\mathcal{M}$ of projectively
complete Hilbert function spaces $\mathcal{H}$, as follows.

######  Theorem 2.27.

Let $\mathcal{H}$ be a projectively complete Hilbert function space on
$\Omega$, let $D\subset\Omega$ and let $\mathcal{M}$ be a local space over
$D$. There exist subspaces $\mathcal{M}_{1},\dots,\mathcal{M}_{n}$ of
$\mathcal{H}$ such that

  * (i)

$\mathcal{M}_{i}$ is punctual for $i=1,\dots,n$;

  * (ii)

$\operatorname{supp}\mathcal{M}_{i}=\\{\alpha_{i}\\}\subset{\rm pc}(D),1\leq
i\leq n$;

  * (iii)

$\mathcal{M}=\mathcal{M}_{1}+\mathcal{M}_{2}+\dots+\mathcal{M}_{n}$;

  * (iv)

$\sum\limits^{n}_{i=1}\dim\mathcal{M}_{i}=\dim\mathcal{M}$, and

  * (v)

$\operatorname{supp}\mathcal{M}=\\{\alpha_{1},\dots,\alpha_{n}\\}$.

In this theorem, $\mathrm{pc}(D)$ denotes the projective closure of a subset
$D$ of $\Omega$, which is defined immediately after Definition 10.11.

##  3\. Examples of pseudomultipliers

In Definition 2.4 we introduced pseudomultipliers $\varphi$ and their
associated multiplication operators $X_{\varphi}$, as a generalization of the
notion of a multiplier (as in Definition 2.1) of a Hilbert function space.
Firstly, we allowed the possibility that $\varphi$ only be defined on a proper
subset of $\Omega$ and secondly we allowed $X_{\varphi}$ only to be defined on
a proper subspace of $\mathcal{H}$ of finite codimension.

Note that, as we assume that $D_{\varphi}$ is a set of uniqueness for
$\mathcal{H}$, if $E_{\varphi}$ and $X_{\varphi}$ are as in Definition 2.4,
then $E_{\varphi}$ is a subspace (not necessarily closed) of $\mathcal{H}$ and
$X_{\varphi}:E_{\varphi}\to\mathcal{H}$ is a linear transformation. In the
case that $E_{\varphi}$ is closed in $\mathcal{H}$, it is an easy consequence
of the closed graph theorem that $X_{\varphi}$ is bounded. Conversely, if
$X_{\varphi}$ is bounded then $E_{\varphi}$ is closed. For suppose $f_{n}\in
E_{\varphi}$ and $f_{n}\to g$ in $\mathcal{H}$. Then the sequence
$(X_{\varphi}f_{n})$ is bounded in $\mathcal{H}$ and, by passing to a
subsequence if necessary, we can suppose that $X_{\varphi}f_{n}$ tends weakly
to some $h\in\mathcal{H}$. For any $\lambda\in D_{\varphi}$ we have

$(X_{\varphi}f_{n})(\lambda)=\varphi(\lambda)f_{n}(\lambda)\to\varphi(\lambda)g(\lambda)$

and

$(X_{\varphi}f_{n})(\lambda)\to h(\lambda).$

Thus $h\in\mathcal{H}$ extends $\varphi(g|D_{\varphi})$, and so $g\in
E_{\varphi}$. Thus $E_{\varphi}$ is closed. In particular, if $E_{\varphi}$
has finite codimension, then $\varphi$ is a pseudomultiplier if and only if
$X_{\varphi}$ is a bounded operator.

There are two ways that a function $\varphi$ defined on a set of uniqueness
can fail to be a pseudomultiplier: $E_{\varphi}$ might not be closed or
$E_{\varphi}$ might be closed but not of finite codimension.

######  Example 3.1.

Let $\Omega=\mathbb{D}$ and let $\mathcal{H}=H^{2}$, the classical Hardy
space. Let $D_{\varphi}=\mathbb{D}$ and define $\varphi$ by
$\varphi(z)=(1-z)^{-1}$. Here $E_{\varphi}=(1-z)H^{2}$ which is a nonclosed
subspace of $H^{2}$. Thus $\varphi$ is not a pseudomultiplier. $\square$

######  Example 3.2.

Let $\Omega=\mathbb{D}^{2}$ and let $\mathcal{H}=H^{2}(\mathbb{D}^{2})$, the
Hardy space on the bidisc. Let
$D_{\varphi}=\\{z\in\mathbb{D}^{2}:z_{1}\not=0\\}$ and define $\varphi$ by
$\varphi(z)=z^{-1}_{1}$. Here $E_{\varphi}=z_{1}H^{2}(\mathbb{D}^{2})$ is a
closed subspace of $\mathcal{H}$. Yet $\varphi$ is not a pseudomultiplier of
$\mathcal{H}$ since $E_{\varphi}$ has infinite codimension. $\square$

There are also two ways that a pseudomultiplier $\varphi$ can fail to be a
multiplier: $D_{\varphi}=\Omega$ but $E_{\varphi}\not=\mathcal{H}$ or
$E_{\varphi}=\mathcal{H}$ but $D_{\varphi}\not=\Omega$.

######  Example 3.3.

Let $\Omega$ be a nonempty open proper subset of $\mathbb{D}$ and let
$\mathcal{H}$ be the space of analytic functions $f$ on $\Omega$ that have an
extension $f^{\sim}$ to $\mathbb{D}$ such that $f^{\sim}\in H^{2}$. Let
$\mathcal{H}$ have the inner product defined by

$\left\langle f,g\right\rangle_{\mathcal{H}}\ =\ \left\langle
f^{\sim},g^{\sim}\right\rangle_{H^{2}}.$ (3.1)

Fix $\alpha\in\mathbb{D}\backslash\Omega$ and define $\varphi$ on $\Omega$ by
$\varphi(z)=(z-\alpha)^{-1}$. Then $D_{\varphi}=\Omega$ and, since equation
(3.1) implies that we may identify $\mathcal{H}$ with $H^{2}$, we have

$E_{\varphi}=\\{f\in\mathcal{H}:f^{\sim}(\alpha)=0\\},$

a closed subspace of $\mathcal{H}$ of codimension 1. Hence $\varphi$ is an
everywhere-defined pseudomultiplier of $\mathcal{H}$ but is not a multiplier
of $\mathcal{H}$. $\square$

######  Example 3.4.

Let $\Omega=\mathbb{D}$ and $\mathcal{H}=H^{2}$. Fix a bounded analytic
function $\psi$ on $\mathbb{D}$ and let $S\subset\mathbb{D}$ be a set of
uniqueness for $\mathcal{H}$. If $\varphi=\psi|S$ then
$E_{\varphi}=\mathcal{H}$ and $\varphi$ is a pseudomultiplier of
$\mathcal{H}$. However, $\varphi$ is not a multiplier unless $S=\mathbb{D}$.

######  Example 3.5.

Let $\Omega_{1}=\mathbb{D}\times\\{1\\},\Omega_{2}=\mathbb{D}\times\\{2\\}$
and $\Omega=\Omega_{1}\cup\Omega_{2}$. Fix a linear transformation $L$ of
$H^{2}$. For $f\in H^{2}$ define $f^{\sim}$ on $\Omega$ by the formula

$f^{\sim}((\lambda,i))=\left\\{\begin{array}[]{ll}f(\lambda)&{\rm if}\qquad
i=1,\\\ \\\ (Lf)(\lambda)&{\rm if}\qquad i=2.\end{array}\right.$

Let $\mathcal{H}=\\{f^{\sim}:f\in H^{2}\\}$ with inner product

$\left\langle f^{\sim},g^{\sim}\right\rangle_{\mathcal{H}}=\left\langle
f,g\right\rangle_{H^{2}}.$ (3.2)

If $\psi$ is a bounded analytic function on $\mathbb{D}$ and $\varphi$ is
defined on $\Omega$ by $\varphi((\lambda,1))=\psi(\lambda)$ then
$E_{\varphi}=\mathcal{H}$ and $\varphi$ is a pseudomultiplier. However,
$\varphi$ is not a multiplier of $\mathcal{H}$ since
$D_{\varphi}=\Omega_{1}\not=\Omega$. $\square$

######  Example 3.6.

A direct sum of pseudomultipliers. In this example we conjoin
pseudomultipliers on two function spaces, $\varphi$ of $\mathcal{H}_{1}$ on a
set $\Omega_{1}$ and $\chi$ of $\mathcal{H}_{2}$ on a set $\Omega_{2}$. Define
$\Omega$ to be the set-theoretic disjoint union $\Omega_{1}\sqcup\Omega_{2}$
of $\Omega_{1}$ and $\Omega_{2}$. Let $\mathcal{H}$ be the direct sum
$\mathcal{H}_{1}\oplus\mathcal{H}_{2}$. Clearly elements of $\mathcal{H}$ are
functions on $\Omega$ in a natural way.

Consider the function $\gamma:D_{\varphi}\sqcup D_{\chi}\to\mathbb{C}$ defined
by

$\gamma(x)=\left\\{\begin{array}[]{lcl}\varphi(x)&\text{ if }&x\in
D_{\varphi}\\\ \chi(x)&\text{ if }&x\in D_{\chi}.\end{array}\right.$

where $D_{\varphi}$ and $D_{\chi}$ are the domains of $\varphi$ and $\chi$,
and so are sets of uniqueness for $\mathcal{H}_{1},\mathcal{H}_{2}$
respectively. For any $\begin{pmatrix}f\\\ g\end{pmatrix}\in\mathcal{H}$, the
function $\gamma\begin{pmatrix}f\\\ g\end{pmatrix}$ on $D_{\varphi}\sqcup
D_{\chi}$ is given by

$\gamma\begin{pmatrix}f\\\
g\end{pmatrix}(x)=\left\\{\begin{array}[]{lcl}\varphi(x)f(x)&\text{ if }&x\in
D_{\varphi}\\\ \chi(x)g(x)&\text{ if }&x\in D_{\chi}.\end{array}\right.$

Now a vector $\begin{pmatrix}f\\\ g\end{pmatrix}\in\mathcal{H}$ satisfies
$\gamma\begin{pmatrix}f\\\ g\end{pmatrix}\in\mathcal{H}$ if and only if
$\varphi f$ extends to an element $h_{1}$ of $\mathcal{H}_{1}$ and $\chi g$
extends to an element $h_{2}$ of $\mathcal{H}_{2}$.

$E_{\gamma}=\left\\{\begin{pmatrix}f\\\ g\end{pmatrix}\in\mathcal{H}:\text{
there exists }h=\begin{pmatrix}h_{1}\\\ h_{2}\end{pmatrix}\in\mathcal{H}\text{
such that }\gamma\begin{pmatrix}f\\\ g\end{pmatrix}=h|(D_{\varphi}\sqcup
D_{\chi})\right\\}$

Hence

$E_{\gamma}=\left\\{\begin{pmatrix}f\\\ g\end{pmatrix}\in\mathcal{H}:f\in
E_{\varphi},g\in E_{\chi}\right\\}=E_{\varphi}\oplus E_{\chi}.$ (3.3)

It is easy to see that if $\dim E_{\varphi}^{\perp}=n$ and $\dim
E_{\chi}^{\perp}=m$, then $E_{\gamma}$ has codimension $n+m$ in $\mathcal{H}$.
Therefore $\gamma$ is a pseudomultiplier of $\mathcal{H}$. $\square$

In Examples 3.3, 3.4 and 3.5, $\Omega$ has been contrived to be either too
large or too small. In other words, $D_{\varphi}$ is too small or too large.
In Example 3.3 we can envisage every $f\in\mathcal{H}$ extending naturally off
$\Omega$ to the larger set $\mathbb{D}$. The function $\varphi$ apparently
“sees” this larger set as well, and though it is bounded on $\Omega$ it fails
to be a multiplier as its values are unbounded near $\alpha$. On the other
hand, in Example 3.4, $\varphi$ becomes a multiplier if we block from our
vision the points of $\mathbb{D}\backslash S$ (that is, replace $\mathcal{H}$
with $H^{2}|S$); alternatively, from its very definition, $\varphi$ has an
extension which is a multiplier of $\mathcal{H}$.

At first glance Example 3.5 resembles Example 3.4: in both we can convert
$\varphi$ into a multiplier by restricting the domain of the Hilbert function
space $\mathcal{H}$. However, there is a qualitative difference: in the former
$\varphi$ has an extension which is a multiplier, in the latter in general it
has not. In Example 3.4 $\varphi$ can somehow see the points outside
$D_{\varphi}$, in Example 3.5 it cannot.

It is clear that if we are to obtain any kind of classification of
pseudomultipliers then we need to address the issue of modification of
domains. A first hope might be that every Hilbert function space has a
“natural” domain of definition; one might then restrict attention to spaces on
their natural domains. After all, it sounds plausible to say that the natural
domain of the restricted Hardy space in Example 3.3 is $\mathbb{D}$. Could
there possibly be a construction analogous to the maximal ideal space of a
commutative Banach algebra? In view of the diversity of examples one can
construct, we think it unlikely. However, in the context of a particular
pseudomultiplier there is a natural notion, which we regard as the most basic
and fruitful concept in the general theory – that of a pseudomultiplier seeing
a vector. Roughly speaking, a pseudomultiplier sees a vector $v$ if $v$ can be
regarded as the kernel function $k_{\lambda}$ of a point $\lambda$ at which
$\varphi$ continues to act as a pseudomultiplier.

We formalized the concept of a pseudomultiplier seeing a vector with value $c$
in Definition 2.7 in the Overview of the paper (Section 2). Recall that a
pseudimultiplier $\varphi$ of $\mathcal{H}$ sees a vector $v\in\mathcal{H}$
with value $c$ if there exists a pseudomultiplier $\psi$ of $\mathcal{H}_{v}$
such that $\psi$ extends $\varphi$, $p\in
D_{\psi},E_{\varphi}=E_{\psi}|\Omega$ and $\psi(p)=c$.

We first probe the meaning of “seeing a vector” in the case of a multiplier
$\varphi$ of $\mathcal{H}$. We can regard $\varphi$ as a pseudomultiplier of
$\mathcal{H}$, with $D_{\varphi}=\Omega$, $E_{\varphi}=\mathcal{H}$, with
singular space $\\{0\\}$ and $X_{\varphi}$ the corresponding operator
$M_{\varphi}:\mathcal{H}\to\mathcal{H}$ of multiplication by $\varphi$.

######  Proposition 3.7.

Let $\varphi$ be a multiplier of a Hilbert function space $\mathcal{H}$ on a
set $\Omega$. The vectors which are visible to $\varphi$ are precisely the
eigenvectors of $M_{\varphi}^{*}$.

###### Proof.

Suppose that $\varphi$ sees $v$ with value $c$. That means that there is a
pseudomultiplier $\psi$ of $\mathcal{H}_{v}$ defined on $\Omega\cup\\{p\\}$
such that $\psi$ extends $\varphi$ and $\psi(p)=c$. Now, for any
$f\in\mathcal{H}$, as functions on $\Omega\cup\\{p\\}$, $\psi f_{v}$ equals
$\varphi f$ on $\Omega$ and equals $c\left\langle f,v\right\rangle$ at $p$.
Hence

$c\left\langle f,v\right\rangle=\left\langle\varphi
f,v\right\rangle=\left\langle f,M_{\varphi}^{*}v\right\rangle.$

Thus $M_{\varphi}^{*}v=\bar{c}v$. The steps are all reversible, so $\varphi$
sees $v$ with value $c$ if and only if $M_{\varphi}^{*}v=\bar{c}v$. $\square$

In particular, for $\lambda\in\Omega$, a multiplier $\varphi$ sees the kernel
$k_{\lambda}$ with value $\varphi(\lambda)$, a fact that we generalized to
pseudomultipliers in Example 2.9.

In the next example $\mathcal{H}\mathcal{H}$ denotes the class of functions on
$\Omega$ which can be expressed as a product $fg$ where $f,g\in\mathcal{H}$.

######  Example 3.8.

The result [5, Theorem 2.1] implies that if $\mathcal{H}$ is a nonsingular
Hilbert function space (in the sense that the vectors
$\\{k_{\lambda}:\lambda\in\Omega\\}$ are linearly independent in
$\mathcal{H}$) and $\varphi$ is a pseudomultiplier of $\mathcal{H}$ with the
property that $D_{\varphi}$ is a set of uniqueness for
$\mathcal{H}\mathcal{H}$, then $\varphi$ sees $k_{\lambda}$ for all
$\lambda\in\Omega\backslash F$ for some set $F\subset\Omega$ containing at
most $\dim(\mathcal{H}\ominus E_{\varphi})$ points.

######  Remark 3.9.

Definition 2.7 allows us to formalize the heuristic remarks made following
Examples 3.3, 3.4, and 3.5. Thus, in Example 3.3, $\varphi$ sees $k_{\lambda}$
with value $(\lambda-\alpha)^{-1}$ for all
$\lambda\in\mathbb{D}\backslash\\{\alpha\\}$ while $\varphi$ does not see
$k_{\alpha}$. On the other hand, in Example 3.4 while strictly speaking
$k_{\lambda}(z)=(1-\overline{\lambda}z)^{-1}$ is the reproducing kernel for
$\mathcal{H}$ only when $\lambda\in\Omega$, $\varphi$ sees $k_{\lambda}$ with
value $\psi(\lambda)$ for all $\lambda\in\mathbb{D}$. In Example 3.5, in
contrast to Example 3.4, $\varphi$ does not necessarily always see the kernel
function. Thus, if $\omega=(\lambda,1)\in\Omega_{1}$, then $\varphi$ sees
$k_{\omega}$ with value $\psi(\lambda)$ but if
$\omega=(\lambda,2)\in\Omega_{2}$, then the reader can easily check that
$\varphi$ sees $k_{\omega}$ with value $c$ if and only if $L^{*}k_{\lambda}$
is an eigenvector for $M^{*}_{\varphi}$ with corresponding eigenvalue
$\overline{c}$, a situation that need not occur.

######  Example 3.10.

In Example 3.3, $\varphi$ does not see $k_{\alpha}$. If we were to drop the
requirement in Definition 2.7 that $E_{\varphi}=E_{\psi}|\Omega$ this would no
longer be true: with the resulting definition, $\varphi$ would see
$k_{\alpha}$. In accordance with Definition 2.7, to determine whether
$\varphi$ sees the vector $v=k_{\alpha}$ in $\mathcal{H}=H^{2}|\Omega$ with
value $c\in\mathbb{C}$ we must consider whether there is a pseudomultiplier
$\psi$ of the Hilbert function space $\mathcal{H}_{v}$ on $\Omega\cup\\{p\\}$
such that $\psi$ extends $\varphi$, $E_{\varphi}=E_{\psi}|\Omega$ and
$\psi(p)=c$. Such a function $\psi$ on $\Omega\cup\\{p\\}$ is necessarily
given by

$\psi(\lambda)=\left\\{\begin{array}[]{cl}\varphi(\lambda)&\qquad\mbox{ for
}\lambda\in\Omega\\\ \\\ c&\qquad\mbox{ if }\lambda=p.\end{array}\right.$

Now $E_{\psi}$ comprises those functions $f_{v}\in\mathcal{H}_{v}$ on
$\Omega\cup\\{p\\}$ such that $\psi f_{v}\in\mathcal{H}_{v}$. By construction
of $\mathcal{H}_{v}$, $\psi f_{v}\in\mathcal{H}_{v}$ if and only if there
exists $g\in H^{2}$ such that, as functions on $\Omega\cup\\{p\\},\ \psi
f_{v}=(g|\Omega)_{v}$, which is to say that $\varphi f=g$ on $\Omega$ and,
corresponding to values at $p$, $c\left\langle f,v\right\rangle=\left\langle
g|\Omega,v\right\rangle.$ Since $v=k_{\alpha}$, the reproducing kernel of
$\alpha$ in $H^{2}$, this pair of equations becomes

$\displaystyle\frac{f(\lambda)}{\lambda-\alpha}$ $\displaystyle=$
$\displaystyle g(\lambda)\mbox{ for all }\lambda\in\Omega$ (3.4)
$\displaystyle cf^{\sim}(\alpha)=c\left\langle f,v\right\rangle=\left\langle
g|\Omega,v\right\rangle$ $\displaystyle=$ $\displaystyle\left\langle
g,k_{\alpha}\right\rangle=g(\alpha)$ (3.5)

(recall that $f^{\sim}$ is the function in $H^{2}$ whose restriction to
$\Omega$ is $f$). From equation (3.4) we can infer that $f^{\sim}(\alpha)=0$
and $g(\alpha)=(f^{\sim})^{\prime}(\alpha)$ which, in conjunction with
equation (3.5) further implies that $g(\alpha)=0$, and therefore
$f^{\sim}(\alpha)=(f^{\sim})^{\prime}(\alpha)=0$. Thus if $f_{v}\in E_{\psi}$
then $f^{\sim}(\alpha)=(f^{\sim})^{\prime}(\alpha)=0$. Conversely, if
$f^{\sim}(\alpha)=(f^{\sim})^{\prime}(\alpha)=0$ then equations (3.4) and
(3.5) hold and so
$E_{\psi}=\\{f_{v}\in\mathcal{H}_{v}:f^{\sim}(\alpha)=(f^{\sim})^{\prime}(\alpha)=0\\}$
and so $E_{\psi}|\Omega$ is a proper subset of $E_{\varphi}$. The present
example explains why we require that $E_{\varphi}=E_{\psi}|\Omega$ in
Definition 2.7. $\square$

######  Lemma 3.11.

Let $\varphi$ be a pseudomultiplier of a Hilbert function space
$(\mathcal{H},\Omega)$, let $v\in\mathcal{H}$ and let $p$ be a point not in
$\Omega$. For any pseudomultiplier $\psi$ of $\mathcal{H}_{v}$ such that
$D_{\psi}\supset D_{\varphi}\cup\\{p\\}$ and $\psi|D_{\varphi}=\varphi$, the
inclusion $E_{\psi}|\Omega\subseteq E_{\varphi}$ holds.

###### Proof.

Consider an arbitrary element of $E_{\psi}|\Omega$: it can be expressed in the
form $f_{v}|\Omega$ for some $f\in\mathcal{H}$. By definition $f_{v}\in
E_{\psi}$ if and only if there exists $g\in\mathcal{H}$ such that $g_{v}=\psi
f_{v}$ on $D_{\psi}$. Evaluating both sides of this equation at any point
$z\in D_{\varphi}$, we find that $g(z)=\varphi(z)f(z)$, and so $f\in
E_{\varphi}$. Thus $E_{\psi}|\Omega\subseteq E_{\varphi}$, as claimed.
$\square$

##  4\. Ambiguous points

In this section we shall repeatedly use a notation introduced in the overview
of the paper. Let $\mathcal{H}$ be a Hilbert function space on $\Omega$. For a
pseudomultiplier $\varphi$ of $\mathcal{H}$, $c\in\mathbb{C}$ and $\alpha\in
D_{\varphi}$, define the pseudomultiplier $\varphi_{c,\alpha}$ on
$D_{\varphi}$ by

$\varphi_{c,\alpha}(\lambda)=\left\\{\begin{array}[]{cl}\varphi(\lambda)&\ \
{\rm if}\quad\lambda\in D_{\varphi}\setminus\\{\alpha\\}\\\ \\\ c&\ \ {\rm
if}\quad\lambda=\alpha.\end{array}\right.$ (4.1)

Recall Definition 2.11: a point $\alpha\in D_{\varphi}$ is an ambiguous point
for  a pseudomultiplier $\varphi$ of $\mathcal{H}$ if there exists
$c\in\mathbb{C}$ with the properties that
$c\not=\varphi(\alpha),E_{\varphi}=E_{\varphi_{c,\alpha}}$, and
$X_{\varphi_{c,\alpha}}=X_{\varphi}$. An ambiguous point is a point at which
the value of a pseudomultiplier $\varphi$ is not determined by the associated
operator $X_{\varphi}$.

In Definition 4.11 below we shall extend the notion of an ambiguous point of
$\varphi$ to points $\alpha\in\Omega$ not assumed to be in $D_{\varphi}$. We
then prove in Theorem 4.14 that $\alpha\in\Omega$ is ambiguous for $\varphi$
if and only if, for every $c\in\mathbb{C},\ \varphi$ sees $k_{\alpha}$ with
value $c$. This linkage between ambiguous points and the notion of seeing a
vector is useful for the organization of the wide variety of examples of
pseudomultipliers and for the analysis of their singularities.

######  Lemma 4.1.

Let $\psi$ be a multiplier of a Hilbert function space $\mathcal{H}$ on a set
$\Omega$. Suppose that $\alpha\in\Omega$ has the property that there exists a
sequence $\\{\lambda_{n}\\}\subseteq\Omega$ with $\lambda_{n}\not=\alpha$ and
$k_{\lambda_{n}}\to k_{\alpha}$. Let $\varphi=\psi_{z,\alpha}$ where
$z\in\mathbb{C}\setminus\\{\psi(\alpha)\\}$. Then $\varphi$ is a
pseudomultiplier of $\mathcal{H}$ and $\alpha$ is an ambiguous point for
$\varphi$.

###### Proof.

This follows immediately from Definition 2.11 and the following simple
observation:

${\rm if}\ c\not=\psi(\alpha),\ {\rm then}\
E_{\psi_{c,\alpha}}=k^{\perp}_{\alpha}\ {\rm and}\
X_{\psi_{c,\alpha}}=X_{\psi}|k^{\perp}_{\alpha}.$ (4.2)

To prove equation (4.2), fix $c\not=\psi(\alpha)$ and first assume that $f\in
k^{\perp}_{\alpha}$. Evidently,
$\psi_{c,\alpha}(\lambda)f(\lambda)=\psi(\lambda)f(\lambda)$ for all
$\lambda\in\Omega\backslash\\{\alpha\\}$. But since $f\in
k^{\perp}_{\alpha},f(\alpha)=0$, and so
$\psi_{c,\alpha}(\alpha)f(\alpha)=\psi(\alpha)f(\alpha)$. Thus, $f\in
k^{\perp}_{\alpha}$ implies that $\psi_{c,\alpha}f=\psi f\in\mathcal{H}$ and
we have shown that $k^{\perp}_{\alpha}\subseteq E_{\psi}$ and
$X_{\psi_{c,\alpha}}|k^{\perp}_{\alpha}=X_{\psi}|k^{\perp}_{\alpha}$. There
remains to show that $E_{\psi_{c,\alpha}}\subseteq k^{\perp}_{\alpha}$.
Accordingly, fix $f\in E_{\psi_{c,\alpha}}$. Thus,
$\psi_{c,\alpha}f\in\mathcal{H}$. But $\psi$ is a multiplier and so we see
that $g=\psi_{c,\alpha}f-\psi f\in\mathcal{H}$. Furthermore, $g(\lambda)=0$ if
$\lambda\not=\alpha$. Recalling our technical hypothesis we see that

$g(\alpha)=\left\langle
g,k_{\alpha}\right\rangle=\lim\limits_{n\to\infty}\left\langle
g,k_{\lambda_{n}}\right\rangle=\lim\limits_{n\to\infty}g(\lambda_{n})=0.$

Hence $0=g(\alpha)=(c-\psi(\alpha))f(\alpha).$ But $c\not=\psi(\alpha)$, so
$f(\alpha)=0$ i.e. $f\in k^{\perp}_{\alpha}$. This concludes the proof of
equation (4.2). $\square$

######  Example 4.2.

Let $\mathcal{H}$ be the space $W^{1,2}[0,1]$ of absolutely continuous
functions on $[0,1]$ whose weak derivatives are in $L^{2}$ and with inner
product given by

$\left\langle
f,g\right\rangle=\int^{1}_{0}f(t)\overline{g(t)}dt+\int^{1}_{0}f^{\prime}(t)\overline{g^{\prime}(t)}dt.$

We showed in Example 2.5 in the introduction that $\chi(t)=\sqrt{t}$ is a
pseudomultiplier of $\mathcal{H}$ with $E_{\chi}=k^{\perp}_{0}$. We claim that
$0$ is an ambiguous point for $\chi$. Indeed,

${\rm if}\ c\in\mathbb{C}\ {\rm is\ arbitrary,\ then}\
E_{\chi_{c,0}}=k^{\perp}_{0}\ {\rm and}\ X_{\chi_{c,0}}=X_{\chi}.$ (4.3)

The proof of the implication (4.3) is very simple. On the one hand, if $f\in
k^{\perp}_{0}$, then $\chi_{c,0}f=\chi f$ so that $f\in E_{\chi_{c,0}}$ and
$X_{\chi_{c,0}}f=X_{\chi}f$. On the other hand, if $f\in E_{\chi_{c,0}}$, then
$\sqrt{t}f\in\mathcal{H}$ which forces $f(0)=0$. $\square$

Observe that there is a qualitative difference between the examples of
ambiguous points implied by Lemma 4.1 and the ambiguous point in Example 4.2.
In Lemma 4.1, $\varphi$ can be redefined at the ambiguous point $\alpha$ in
such a way that $\alpha$ is no longer an ambiguous point. But in Example 4.2,
$0$ remains an ambiguous point for $\chi$, regardless of how $\chi$ is
redefined at $0$ (this follows from the implication (4.3)). We formalize these
two cases in the following definition.

######  Definition 4.3.

Let $\varphi$ be a pseudomultiplier of $\mathcal{H}$ and let $\alpha\in
D_{\varphi}$ be an ambiguous point for $\varphi$. We say $\alpha$ is a
removable ambiguous point for $\varphi$ if there exists $c\in\mathbb{C}$ such
that $\alpha$ is not an ambiguous point for $\varphi_{c,\alpha}$.  We say
$\alpha$ is an essential ambiguous point for $\varphi$ if $\alpha$ is an
ambiguous point but not a removable ambiguous point for $\varphi$.

Thus, in Lemma 4.1 $\alpha$ is a removable ambiguity while in Example 4.2,
$\alpha$ is an essential ambiguity. Before continuing our investigation of
ambiguous points we state and prove some useful technical lemmas.

######  Lemma 4.4.

If $\varphi$ is a pseudomultiplier, $\alpha\in D_{\varphi}$ and
$E_{\varphi}\subset k^{\perp}_{\alpha}$ then, for any
$c\in\mathbb{C},E_{\varphi}=E_{\varphi_{c,\alpha}}\cap k^{\perp}_{\alpha}$ and
$X_{\varphi}=X_{\varphi_{c,\alpha}}|E_{\varphi}$.

###### Proof.

If $f\in E_{\varphi}$ then $f(\alpha)=0$ and there exists $h\in\mathcal{H}$
such that $h=f\varphi$ on $D_{\varphi}$. Hence $h=\varphi
f=\varphi_{c,\alpha}f$ on $D_{\varphi}$, and so $f\in E_{\varphi_{c,\alpha}}$.
Thus $E_{\varphi}\subseteq E_{\varphi_{c,\alpha}}\cap k^{\perp}_{\alpha}$ and
$X_{\varphi}=X_{\varphi_{c,\alpha}}|E_{\varphi}$. Conversely, assume that
$f\in E_{\varphi_{c,\alpha}}\cap k^{\perp}_{\alpha}$. Then $f(\alpha)=0$ and
there exists $h\in\mathcal{H}$ such that $\varphi_{c,\alpha}f=h$ on
$D_{\varphi}$. Hence $\varphi f=\varphi_{c,\alpha}f=h$ on $D_{\varphi}$ and so
$f\in E_{\varphi}$. Thus $E_{\varphi_{c,\alpha}}\cap
k^{\perp}_{\alpha}\subseteq E_{\varphi}$. $\square$

######  Lemma 4.5.

If $\varphi$ is a pseudomultiplier, $\alpha\in D_{\varphi}$ and
$E_{\varphi}\subseteq k^{\perp}_{\alpha}$, then there can exist at most one
complex number $c$ such that $E_{\varphi_{c,\alpha}}$ is not a subset of
$k^{\perp}_{\alpha}$.

###### Proof.

Let $\varphi$ and $\alpha$ satisfy the hypotheses of the lemma and assume that
$c_{1},c_{2}\in\mathbb{C}$ with $E_{\varphi_{c_{1},\alpha}}\not\subseteq
k^{\perp}_{\alpha}$ and $E_{\varphi_{c_{2},\alpha}}\not\subseteq
k^{\perp}_{\alpha}$. We need to prove that $c_{1}=c_{2}$. Since
$E_{\varphi_{c_{1},\alpha}}\not\subseteq k^{\perp}_{\alpha}$, there exists
$f_{1}\in E_{\varphi_{c_{1},\alpha}}$ with $f_{1}(\alpha)=1$. Similarly, pick
$f_{2}\in E_{\varphi_{c_{2},\alpha}}$ with $f_{2}(\alpha)=1$. Since $f_{1}\in
E_{\varphi_{c_{1},\alpha}}$, there exists $h_{1}\in\mathcal{H}$ such that
$\varphi_{c_{1},\alpha}f_{1}=h_{1}$ on $D_{\varphi}$. Similarly, there exists
$h_{2}\in\mathcal{H}$ such that $\varphi_{c_{2},\alpha}f_{2}=h_{2}$ on
$D_{\varphi}$. Note that since
$f_{1}(\alpha)=f_{2}(\alpha)=1,\varphi_{c_{1},\alpha}(\alpha)=c_{1}$ and
$\varphi_{c_{2},\alpha}(\alpha)=c_{2}$, we have $h_{1}(\alpha)=c_{1}$ and
$h_{2}(\alpha)=c_{2}$. Now define $f\in\mathcal{H}$ by the formula

$f=(c_{2}-\varphi(\alpha))f_{1}-(c_{1}-\varphi(\alpha))f_{2},$

and define $h\in\mathcal{H}$ by the formula

$h=(c_{2}-\varphi(\alpha))h_{1}-(c_{1}-\varphi(\alpha))h_{2}.$

We claim that $\varphi f=h$ on $D_{\varphi}$. To see this, first fix
$\lambda\in D_{\varphi}\backslash\\{\alpha\\}$ and note that

$\displaystyle\varphi(\lambda)f(\lambda)$ $\displaystyle=$
$\displaystyle(c_{2}-\varphi(\alpha))\varphi(\lambda)f_{1}(\lambda)-(c_{1}-\varphi(\alpha))\varphi(\lambda)f_{2}(\lambda)$
$\displaystyle=$
$\displaystyle(c_{2}-\varphi(\alpha))\varphi_{c_{1},\alpha}(\lambda)f_{1}(\lambda)-(c_{1}-\varphi(\alpha))\varphi_{c_{2},\alpha}((\lambda)f_{2}(\lambda)$
$\displaystyle=$
$\displaystyle(c_{2}-\varphi(\alpha))h_{1}(\lambda)-(c_{1}-\varphi(\alpha))h_{2}(\lambda)$
$\displaystyle=$ $\displaystyle h(\lambda).$

On the other hand,

$\displaystyle\varphi(\alpha)f(\alpha)$ $\displaystyle=$
$\displaystyle\varphi(\alpha)\Big{(}(c_{2}-\varphi(\alpha)f_{1}(\alpha)-(c_{1}-\varphi(\alpha))f_{2}(\alpha)\Big{)}$
$\displaystyle=$
$\displaystyle\varphi(\alpha)\Big{(}(c_{2}-\varphi(\alpha))-(c_{1}-\varphi(\alpha))\Big{)}$
$\displaystyle=$ $\displaystyle\varphi(\alpha)(c_{2}-c_{1})$ $\displaystyle=$
$\displaystyle\Big{(}c_{2}-\varphi(\alpha)\Big{)}c_{1}-\Big{(}c_{1}-\varphi(\alpha)\Big{)}c_{2}$
$\displaystyle=$
$\displaystyle\Big{(}c_{2}-\varphi(\alpha)\Big{)}h_{1}(\alpha)-\Big{(}c_{1}-\varphi(\alpha)\Big{)}h_{2}(\alpha)$
$\displaystyle=$ $\displaystyle h(\alpha).$

Thus, in fact, $\varphi f=h$ on $D_{\varphi}$. But this means that $f\in
E_{\varphi}$. Since, by assumption, $E_{\varphi}\subseteq
k^{\perp}_{\alpha},f\in k^{\perp}_{\alpha}$, i.e. $f(\alpha)=0$. Thus

$\displaystyle 0$ $\displaystyle=$ $\displaystyle f(\alpha)$ $\displaystyle=$
$\displaystyle\Big{(}c_{2}-\varphi(\alpha)\Big{)}f_{1}(\alpha)-\Big{(}c_{1}-\varphi(\alpha)\Big{)}$
$\displaystyle=$
$\displaystyle\Big{(}c_{2}-\varphi(\alpha)\Big{)}-\Big{(}c_{1}-\varphi(\alpha)\Big{)}$
$\displaystyle=$ $\displaystyle c_{2}-\ c_{1},$

so that $c_{1}=c_{2}$ as was to be shown. $\square$

Lemmas 4.4 and 4.5 combine to give a simple criterion for a point $\alpha$ to
be an ambiguous point for a pseudomultiplier.

######  Proposition 4.6.

Let $\varphi$ be a pseudomultiplier and let $\alpha\in D_{\varphi}$. Then
$\alpha$ is an ambiguous point for $\varphi$ if and only if
$E_{\varphi}\subseteq k^{\perp}_{\alpha}$.

###### Proof.

First assume that $E_{\varphi}\subseteq k^{\perp}_{\alpha}$. By Lemma 4.5,
there exists $c\not=\varphi(\alpha)$ such that
$E_{\varphi_{c,\alpha}}\subseteq k^{\perp}_{\alpha}$. But then Lemma 4.8
implies that $E_{\varphi}=E_{\varphi_{c,\alpha}}$ and
$X_{\varphi}=X_{\varphi_{c,\alpha}}$. Thus $\alpha$ is an ambiguous point for
$\varphi$.

Conversely, assume that $\alpha$ is an ambiguous point for $\varphi$. Thus
there exists $c\not=\varphi(\alpha)$ such that
$E_{\varphi}=E_{\varphi_{c,\alpha}}$ and $X_{\varphi}=X_{\varphi_{c,\alpha}}$.
We need to show that $E_{\varphi}\subseteq k^{\perp}_{\alpha}$. Accordingly,
fix $f\in E_{\varphi}$. To see that $f\in k^{\perp}_{\alpha}$ we argue by
contradiction. Assume that $f(\alpha)\not=0$. Now since $f\in
E_{\varphi}=E_{\varphi_{c,\alpha}}$ there exist $h_{1},h_{2}\in\mathcal{H}$
such that $\varphi f=h_{1}$ and $\varphi_{c,\alpha}f=h_{2}$ on $D_{\varphi}$.
Hence, if we set $h=h_{1}-h_{2}$ we have that $h\in\mathcal{H}$ and

$\displaystyle h(\lambda)$ $\displaystyle=$ $\displaystyle
h_{1}(\lambda)-h_{2}(\lambda)$ $\displaystyle=$
$\displaystyle\varphi(\lambda)f(\lambda)-\varphi_{c,\alpha}(\lambda)f(\lambda)$
$\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{cl}0&{\rm
if}\quad\lambda\in D_{\varphi}\backslash\\{\alpha\\}\\\ \\\
\Big{(}\varphi(\alpha)-c\Big{)}f(\alpha)&{\rm
if}\quad\lambda=\alpha.\end{array}\right.$

Note that since $\varphi(\alpha)\not=c$ and $f(\alpha)\not=0$, this formula
implies that $h\not=0$. Also, the formula implies that $\varphi
h=\varphi(\alpha)h$ on $D_{\varphi}$ so that $h\in E_{\varphi}$ and
$X_{\varphi}h=\varphi(\alpha)h$. But the formula similarly implies that
$X_{\varphi_{c,\alpha}}h=ch$. Thus, since $\varphi(\alpha)\not=c$ and
$h\not=0,X_{\varphi_{c,\alpha}}h\not=X_{\varphi}h$, contradicting the fact
that $X_{\varphi}=X_{\varphi_{c,\alpha}}$. This completes the proof of
Proposition 4.6. $\square$

Thus $\alpha$ is an ambiguous point for $\varphi$ if and only if
$E_{\varphi}\subseteq k^{\perp}_{\alpha}$. A consequence of Lemmas 4.4 and 4.5
is that, for ambiguous points $\alpha$, there is a rather rigid dichotomy for
the behavior of the spaces $E_{\varphi_{c,\alpha}}$ as $c$ varies.

######  Lemma 4.7.

Let $\varphi$ be a pseudomultiplier and let $\alpha\in D_{\varphi}$. If
$\alpha$ is an ambiguous point for $\varphi$ then exactly one of the following
two statements holds:

  1. (i)

$E_{\varphi_{c,\alpha}}\subseteq k^{\perp}_{\alpha}\text{ for all
}c\in\mathbb{C};$ (4.5)

  2. (ii)

There exists $\gamma\in\mathbb{C}$ such that

$E_{\varphi_{\gamma,\alpha}}\not\subset k^{\perp}_{\alpha}\text{ but
}E_{\varphi_{c,\alpha}}\subset k^{\perp}_{\alpha}$ (4.6)

for all $c\in\mathbb{C}\smallsetminus\\{\gamma\\}$.

Furthermore,

if equation (4.5) holds then $E_{\varphi_{c,\alpha}}=E_{\varphi}$ and
$X_{\varphi_{c,\alpha}}=X_{\varphi}$ for all $c\in\mathbb{C}$;

if equation (4.6) holds then $E_{\varphi_{c,\alpha}}=E_{\varphi}$ and
$X_{\varphi_{c,\alpha}}=X_{\varphi}$ for all
$c\in\mathbb{C}\smallsetminus\\{\gamma\\}$ while $E_{\varphi}$ is a subspace
of $E_{\varphi_{\gamma,\alpha}}$ of codimension 1 and
$X_{\varphi}=X_{\varphi_{\gamma,\alpha}}|E_{\varphi}$.

###### Proof.

That either of equations (4.5) or (4.6) occurs is simply a restatement of
Lemma 4.5. If (4.5) occurs, then Lemma 4.4 implies that
$E_{\varphi_{c,\alpha}}=E_{\varphi}$ and $X_{\varphi_{c,\alpha}}=X_{\varphi}$
for all $c\in\mathbb{C}$. If (4.6) occurs then again Lemma 4.4 implies that
$E_{\varphi_{c,\alpha}}=E_{\varphi}$ and $X_{\varphi_{c,\alpha}}=X_{\varphi}$
for all $c\in\mathbb{C}\backslash\\{\gamma\\}$. Finally, suppose that (4.6)
occurs, so that $E_{\varphi_{\gamma,\alpha}}\not\subseteq k^{\perp}_{\alpha}$.
Since by Lemma 4.4, $E_{\varphi}=E_{\varphi_{c,\alpha}}\cap
k^{\perp}_{\alpha}$, necessarily $E_{\varphi}\subseteq
E_{\varphi_{\gamma,\alpha}}$ and $E_{\varphi}$ has codimension 1 in
$E_{\varphi_{\gamma,\alpha}}$. Also, in this case, Lemma 4.4 implies that
$X_{\varphi}=X_{\varphi_{\gamma,\alpha}}|E_{\varphi}$. $\square$

Lemma 4.7 has a number of corollaries about the nature of ambiguous points.
The key observation is that in Lemma 4.7, the inclusion (4.5) occurs when
$\alpha$ is essential, and the relation (4.6) occurs when $\alpha$ is
removable.

######  Proposition 4.8.

Let $\varphi$ be a pseudomultiplier and assume that $\alpha\in D_{\varphi}$ is
an ambiguous point for $\varphi$. Then $\alpha$ is an essential ambiguous
point for $\varphi$ if and only if $E_{\varphi_{c,\alpha}}\subset
k^{\perp}_{\alpha}$ for all $c\in\mathbb{C}$.

###### Proof.

By definition, $\alpha$ is essential if and only if $\alpha$ is ambiguous for
$\varphi_{c,\alpha}$ for all $c\in\mathbb{C}$. Hence the proposition follows
immediately from Proposition 4.6. $\square$

Notice that in Definition 2.11 $\alpha$ was said to be ambiguous if $\varphi$
could be redefined at $\alpha$ without the underlying operator $X_{\varphi}$
changing. In fact, if $\alpha$ is ambiguous, $\varphi$ can be arbitrarily
redefined at $\alpha$ if $\alpha$ is essential and arbitrarily redefined at
$\alpha$ with one exception if $\alpha$ is removable.

######  Proposition 4.9.

Let $\varphi$ be a pseudomultiplier and let $\alpha\in D_{\varphi}$ be an
ambiguous point for $\varphi$.

  1. (i)

If $\alpha$ is essential, then $E_{\varphi_{c,\alpha}}=E_{\varphi}$ and
$X_{\varphi_{c,\alpha}}=X_{\varphi}$ for all $c\in\mathbb{C}$.

  2. (ii)

If $\alpha$ is removable then there exists a unique $\gamma\in\mathbb{C}$ such
that $E_{\varphi_{c,\alpha}}=E_{\varphi}$ if and only if
$c\in\mathbb{C}\backslash\\{\gamma\\}$. Moreover, this number $\gamma$ has the
property that $X_{\varphi_{c,\alpha}}=X_{\varphi}$ if and only if
$c\in\mathbb{C}\backslash\\{\gamma\\}$.

###### Proof.

(i) Suppose that $\alpha$ is an essential ambiguous point for $\varphi$ and
$c\in\mathbb{C}$. By Proposition 4.6 $E_{\varphi}\subseteq
k_{\alpha}^{\perp}$. Thus, for any function $f\in E_{\varphi}$, $f(\alpha)=0$
and so $\varphi_{c,\alpha}f=\varphi f$ on $D_{\varphi}$, from which it follows
that $E_{\varphi_{c,\alpha}}=E_{\varphi}$ and
$X_{\varphi_{c,\alpha}}=X_{\varphi}$.

(ii) Suppose that $\alpha$ is a removable ambiguous point for $\varphi$. Then
$\alpha$ is not an essential ambiguous point for $\varphi$, and so, by
Proposition 4.8, there exists $\gamma\in\mathbb{C}$ such that
$E_{\varphi_{\gamma,\alpha}}\not\subset k_{\alpha}^{\perp}$. Thus, in the
dichotomy described in Lemma 4.7 it is alternative (ii) that holds, and so
$E_{\varphi_{c,\alpha}}\subset k_{\alpha}^{\perp}$ for all
$c\in\mathbb{C}\setminus\\{\gamma\\}$, and by the “Furthermore” of Lemma 4.7,
$E_{\varphi_{c,\alpha}}=E_{\varphi}$ and $X_{\varphi_{c,\alpha}}=X_{\varphi}$
for all $c\in\mathbb{C}\setminus\\{\gamma\\}$. It follows that $\gamma$ is
unique. $\square$

We conclude our remarks concerning ambiguous points $\alpha\in D_{\varphi}$ by
noting the following fact.

######  Proposition 4.10.

Let $\varphi$ be a pseudomultiplier on $\mathcal{H}$ and let $\alpha\in
D_{\varphi}$. If $\alpha$ is a removable ambiguous point for $\varphi$ then

  1. (i)

there exists a unique $c\in\mathbb{C}$ such that $\alpha$ is not an ambiguous
point for $\varphi_{c,\alpha}$,

  2. (ii)

there exists a unique $d\in\mathbb{C}$ such that $E_{\varphi}\subseteq
E_{\varphi_{d,\alpha}}$. Finally, $c=d$ and
$X_{\varphi}=X_{\varphi_{c,\alpha}}|E_{\varphi}$.

###### Proof.

Suppose that $\alpha$ is a removable ambiguous point for $\varphi$.

(i) By Definition 4.3, there exists $c\in\mathbb{C}$ such that $\alpha$ is not
an ambiguous point for $\varphi_{c,\alpha}$, so that, to prove statement (i),
we must only show that this $c$ is unique. But by Proposition 4.9(ii), this
value (denoted by $\gamma$) is unique. So statement (i) holds.

(ii) Since $\alpha$ is a removable ambiguous point for $\varphi$, $\alpha$ is
not an essential ambiguous point for $\varphi$, and so, by Proposition 4.8, we
may choose $d\in\mathbb{C}$ such that $E_{\varphi_{d,\alpha}}\not\subset
k_{\alpha}^{\perp}$. Thus, in the dichotomy described in Lemma 4.7 it is
alternative (ii) that holds, with $\gamma=d$, and so
$E_{\varphi_{z,\alpha}}\subset k_{\alpha}^{\perp}$ for all
$z\in\mathbb{C}\setminus\\{d\\}$, $E_{\varphi}$ is a subspace of
$E_{\varphi_{d,\alpha}}$ of codimension 1 and
$X_{\varphi}=X_{\varphi_{d,\alpha}}|E_{\varphi}$. We claim that this $d$ is
unique. But by Proposition 4.9(ii), this value (denoted by $\gamma$) is
unique. It is clear that $\alpha$ is not an ambiguous point for
$\varphi_{d,\alpha}$, and so, by the uniqueness statement in part (i), $c=d$.
Thus statement (ii) of Proposition 4.10 holds. $\square$

The concept of an ambiguous point can be extended naturally to points not
assumed to be in the domain of $\varphi$. Recall Example 2.9, if $\varphi$ is
a pseudomultiplier of a Hilbert function space $\mathcal{H}$ on $\Omega$ and
if $\lambda\in D_{\varphi}$, then $\varphi$ sees $k_{\lambda}$ with value
$\varphi(\lambda)$. Let us combine the definition of an ambiguous point
$\alpha\in D_{\varphi}$ with the notion of $\varphi$ seeing $k_{\alpha}$ to
yield a concept of an ambiguous point $\alpha\in\Omega\setminus D_{\varphi}$.

######  Definition 4.11.

Let $\varphi$ be a pseudomultiplier and let $\alpha\in\Omega\backslash
D_{\varphi}$. We say $\alpha$ is an ambiguous point for $\varphi$ if there
exists $c\in\mathbb{C}$ such that $\varphi$ sees $k_{\alpha}$ with value $c$
and $\alpha$ is an ambiguous point for the pseudomultiplier $\psi_{c,\alpha}$
defined on $D_{\varphi}\cup\\{\alpha\\}$ by the formula

$\psi_{c,\alpha}(\lambda)=\left\\{\begin{array}[]{cl}\varphi(\lambda)&\mbox{
if }\lambda\in D_{\varphi}\\\ \\\ c&\mbox{ if
}\lambda=\alpha\end{array}\right..$ (4.7)

######  Proposition 4.12.

Let $\varphi$ be a pseudomultiplier of $\mathcal{H}$ on $\Omega$ and let
$\alpha\in\Omega\setminus D_{\varphi}$. If $\alpha$ is an ambiguous point of
$\varphi$ then $\alpha$ is an essential ambiguous point for $\psi_{c,\alpha}$
as defined in equation (4.7).

###### Proof.

By the definition of ambiguous point of $\varphi$, there is $c\in\mathbb{C}$
such that $\varphi$ sees $k_{\alpha}$ with value $c$ and
$E_{\varphi}=E_{\psi_{c,\alpha}}$. Also by definition, $\alpha$ is ambiguous
for $\psi_{c,\alpha},$ and so, by Proposition 4.6,
$E_{\psi_{c,\alpha}}\subseteq k^{\perp}_{\alpha}$, and therefore
$E_{\varphi}\subseteq k^{\perp}_{\alpha}$. We claim that, for every
$d\in\mathbb{C}$, $\alpha$ is an ambiguous point for $\varphi_{d,\alpha}$. For
all $d\in\mathbb{C},\psi_{d,\alpha}$ extends $\varphi$, and so
$E_{\psi_{d,\alpha}}\subseteq E_{\varphi}$. Thus, for all
$d\in\mathbb{C},E_{\psi_{d,\alpha}}\subseteq k_{\alpha}^{\perp}$. Therefore,
for all
$d\in\mathbb{C},E_{(\psi_{c,\alpha})_{d,\alpha}}=E_{\psi_{d,\alpha}}\subseteq
k^{\perp}_{\alpha}$. By Proposition 4.8 $\alpha$ is essential for
$\psi_{c,\alpha}$. $\square$

In light of Proposition 4.12, the following analog to Proposition 4.9 is not a
surprise.

######  Proposition 4.13.

Let $\varphi$ be a pseudomultiplier and let $\alpha\in\Omega\smallsetminus
D_{\varphi}$. For $c\in\mathbb{C}$ let $\psi_{c,\alpha}$ be defined by
equation (4.7). Then $\alpha$ is an ambiguous point for $\varphi$ if and only
if, for all $c\in\mathbb{C},E_{\psi_{c,\alpha}}=E_{\varphi}$ and
$X_{\psi_{c,\alpha}}=X_{\varphi}$.

We conclude the section with a result which gives a succinct characterization
of ambiguity applicable to both cases, $\alpha\in D_{\varphi}$ and
$\alpha\in\Omega\smallsetminus D_{\varphi}$.

######  Theorem 4.14.

Let $\varphi$ be a pseudomultiplier and let $\alpha\in\Omega$. Then $\alpha$
is ambiguous for $\varphi$ if and only if, for every $c\in\mathbb{C},\
\varphi$ sees $k_{\alpha}$ with value $c$.

###### Proof.

First suppose that $\varphi$ sees $k_{\alpha}$ with arbitrary value. Thus, for
each $c$ there exists a pseudomultiplier $\omega_{c}$ of
$\mathcal{H}_{k_{\alpha}}$ such that $\omega_{c}$ extends
$\varphi,E_{\varphi}=E_{\omega_{c}}|\Omega,p\in D_{\omega_{c}}$, and
$\omega_{e}(p)=c$. In particular if $f\in E_{\varphi}$ then $f_{k_{\alpha}}\in
E_{\omega_{c}}$ and there exists $h\in\mathcal{H}$ such that
$\omega_{c}f_{k}=h_{k_{\alpha}}$. Noting that $\omega_{c}$ extends $\varphi$
and $\omega_{c}(p)=c$, we deduce that the following holds.

If $f\in E_{\varphi}$, then there exists $h\in\mathcal{H}$ such that

$\displaystyle\varphi(\lambda)f(\lambda)$ $\displaystyle=$ $\displaystyle
h(\lambda)\qquad{\rm if}\ \lambda\in D_{\varphi}\quad{\rm and}$ (4.8)
$\displaystyle c\ f(\alpha)$ $\displaystyle=$ $\displaystyle h(\alpha)$

There are two possibilities: either $\alpha\in D_{\varphi}$ or
$\alpha\in\Omega\smallsetminus D_{\varphi}$. If $\alpha\in D_{\varphi}$, then
noting that in equations (4.8), while $h$ depends on $c,\varphi$ and $f$ do
not, we deduce that $\varphi(\alpha)f(\alpha)=cf(\alpha)$ for all $c$. Hence,
$f(\alpha)=0$ if $f\in E_{\varphi}$. Proposition 4.6 implies that $\alpha$ is
an ambiguous point for $\varphi$.

Now assume that $\alpha$ is an ambiguous point for $\varphi$. Fix $c$ and
define $\omega_{c}$ on $D_{\varphi}\cup\\{p\\}\subseteq\Omega\cup\\{p\\}$ by
the formula

$\omega_{c}(\lambda)=\left\\{\begin{array}[]{cl}\varphi(\lambda)&{\rm
if}\quad\lambda\in D_{\varphi}\\\ \\\ c&{\rm
if}\quad\lambda=p\end{array}\right.$

Then $\omega_{c}$ is a pseudomultiplier, $\omega_{c}$ extends
$\varphi,\omega_{c}(p)=c$, and $E_{\omega_{c}}|\Omega\subseteq E_{\varphi}$.
Hence $\varphi$ will see $k_{\alpha}$ with value $c$ if we can show that $f\in
E_{\varphi}$ implies that $f_{k_{\alpha}}\in E_{\omega_{c}}$. A moment’s
reflection revelas that this last condition is equivalent to (4.8). Because
$\alpha$ is an ambiguous point for $\varphi$, it is easy to check that
equations (4.8) follow from Proposition 4.6 when $\alpha\in D_{\varphi}$ and
follow from Proposition 4.13 when $\alpha\in\Omega\smallsetminus D_{\varphi}$.
This concludes the proof of Theorem 4.14. $\square$

##  5\. Ambiguous vectors and the ambiguous space

We have shown that a point $\alpha\in\Omega$ is an ambiguous point for a
pseudomultiplier $\varphi$ if and only if $\varphi$ sees $k_{\alpha}$ with
arbitrary value. We have also abstracted this property to say that a vector
$v\in\mathcal{H}$ is an ambiguous vector for a pseudomultiplier $\varphi$ of
$\mathcal{H}$ if $\varphi$ sees $v$ with arbitrary value (see Definition
2.13).  Thus, if $\varphi$ is a pseudomultiplier and $\alpha$ is a point,
$\alpha$ is an ambiguous point for $\varphi$ if and only if $k_{\alpha}$ is an
ambiguous vector for $\varphi$. An ambiguous vector need not be a kernel. In
this section we shall study general ambiguous vectors for $\varphi$.

The following lemma gives an operator-theoretic expression of the concepts of
seeing a vector and ambiguity. Recall that if $\varphi$ is a pseudomultiplier
then $X_{\varphi}$ is a bounded linear transformation from $E_{\varphi}$ into
$\mathcal{H}$. Thus $X^{*}_{\varphi}$ is a bounded linear transformation from
$\mathcal{H}$ into $E_{\varphi}$. Recall also a basic fact in the theory of
multipliers that if $\varphi$ is a multiplier of $\mathcal{H}$ and
$k_{\lambda}$ is the reproducing kernel for $\lambda\in\Omega$, then
$X_{\varphi}^{*}k_{\lambda}=\overline{\varphi(\lambda)}k_{\lambda}$. We
generalize this fact to pseudomultipliers in the following lemma.

######  Lemma 5.1.

Let $\varphi$ be a pseudomultiplier of $\mathcal{H}$ and let
$v\in\mathcal{H}$. Then

(i) $\varphi$ sees $v$ with value $c$ if and only if there exists $u\in
E^{\perp}_{\varphi}$ such that

$X^{*}_{\varphi}v=\overline{c}v+u.$ (5.1)

(ii) $\varphi$ sees $v$ with value $c$ if and only if

$X^{*}_{\varphi}v=\overline{c}P_{E_{\varphi}}v.$ (5.2)

###### Proof.

(i). Fix a pseudomultiplier $\varphi,c\in\mathbb{C}$, and $v\in\mathcal{H}$.
Define $\psi$ on $D_{\varphi}\cup\\{p\\}\subseteq\Omega\cup\\{p\\}$ by the
formula

$\psi(\lambda)=\left\\{\begin{array}[]{cc}\varphi(\lambda)&\qquad\mbox{ for
all }\lambda\in D_{\varphi}\\\ \\\ c&\qquad\text{ if
}\lambda=p,\end{array}\right.$

so that $\varphi$ sees $v$ with value $c$ if and only if

$(E_{\psi})|\Omega=E_{\varphi}.$ (5.3)

By Lemma 3.11, equation (5.3) is equivalent to the inclusion

$(E_{\psi})|\Omega\supset E_{\varphi}.$ (5.4)

Assume that $\varphi$ sees $v$ with value $c$, so that equation (5.4) holds.
It means that, for any $f\in E_{\varphi}$ there exists $g\in\mathcal{H}$ such
that $g_{v}=\psi f_{v}$ on $D_{\psi}$. It follows that
$g(z)=\varphi(z)f(z)=(X_{\varphi}f)(z)$ for all $z\in D_{\varphi}$, and (since
$g_{v}(p)=(\psi f)(p)$)

$c\left\langle f,v\right\rangle=\left\langle g,v\right\rangle.$ (5.5)

Thus

$c\left\langle f,v\right\rangle=\left\langle
X_{\varphi}f,v\right\rangle\qquad{\rm for\ all}\quad f\in E_{\varphi},$

and so $X_{\varphi}^{*}v-\bar{c}v\in E_{\varphi}^{\perp}$, and equation (5.1)
holds.

Conversely, suppose there exists $u\in E^{\perp}_{\varphi}$ such that
$X^{*}_{\varphi}v=\overline{c}v+u$. Thus

$\left\langle f,X^{*}_{\varphi}v\right\rangle=\left\langle
f,\overline{c}v\right\rangle\qquad{\rm for\ all}\quad f\in E_{\varphi},$

from which it follows that

$\left\langle X_{\varphi}f,v\right\rangle=c\left\langle
f,v\right\rangle\qquad{\rm for\ all}\quad f\in E_{\varphi},$

and hence

$\left(X_{\varphi}f\right)_{v}(p)=cf_{v}(p)\qquad{\rm for\ all}\quad f\in
E_{\varphi}\ .$ (5.6)

We claim that equation (5.3) holds. To see this, consider any $f\in
E_{\varphi}$. To show that $f\in E_{\psi}|\Omega$ it suffices to prove that
$f_{v}\in E_{\psi}$, since $f_{v}|\Omega=f$. Now $f_{v}\in E_{\psi}$ if and
only if there exists $g\in\mathcal{H}$ such that $\psi f_{v}=g_{v}$, which is
to say that $\varphi f=g$ on $D_{\varphi}$, and, corresponding to evaluation
at $p$, $c\left\langle f,v\right\rangle=\left\langle g,v\right\rangle$.
Equation (5.6) shows that we may obtain such an element $g$ of $\mathcal{H}$
by choosing $g=X_{\varphi}f$. We have shown that $\varphi$ sees $v$ with value
$c$. This establishes Lemma 5.1 (i).

(ii) is an immediate consequence of Part (i) and the fact that ${\rm ran}\
X^{*}_{\varphi}\subseteq E_{\varphi}$. $\square$

######  Corollary 5.2.

Scaling of visible vectors. If $\varphi$ sees a vector $v$ with value $c$
then, for any $\lambda\in\mathbb{C}$, $\varphi$ sees $\lambda v$ with value
$\bar{\lambda}c$.

###### Proof.

Suppose that $\varphi$ sees a vector $v\in\mathcal{H}$ with value $c$ and
consider any $\lambda\in\mathbb{C}$. By Lemma 5.1 there exists $u\in
E^{\perp}_{\varphi}$ such that $X^{*}_{\varphi}v=\overline{c}v+u$. Multiply
this equation through by $\lambda$ to get

$X^{*}_{\varphi}\lambda v=\overline{c}\lambda v+\lambda u.$

Since $\lambda u\in E^{\perp}_{\varphi}$, it follows from Lemma 5.1 that
$\varphi$ sees $\lambda v$ with value $\bar{\lambda}c$. $\square$

The set of ambiguous vectors of a pseudomultiplier $\varphi$ plays an
important role: recall that in Definition 2.13 we denoted it by
$\mathcal{A}_{\varphi}$. As a further corollary of Lemma 5.1 we obtain an
operator-theoretic characterization of ambiguous vectors and the ambiguous
space.

######  Proposition 5.3.

Let $\varphi$ be a pseudomultiplier and let $v\in\mathcal{H}$. Then

(i) $v$ is an ambiguous vector for $\varphi$ if and only if $v\in
E^{\perp}_{\varphi}\cap\ker X^{*}_{\varphi}$.

(ii)

$\mathcal{A}_{\varphi}=E^{\perp}_{\varphi}\cap\ker X^{*}_{\varphi}$ (5.7)

is a finite-dimensional subspace of $\mathcal{H}$.

###### Proof.

(i). Suppose $v$ is ambiguous for $\varphi$. By Lemma 5.1,

$X^{*}_{\varphi}v=\overline{c}P_{E_{\varphi}}v$

for every $c\in\mathbb{C}$. Taking $c=0$ gives $v\in\ker X^{*}_{\varphi}$ and
taking $c=1$ gives $v\in E^{\perp}_{\varphi}$. Conversely, if $v\in
E^{\perp}_{\varphi}\cap\ker X^{*}_{\varphi}$ then both sides of equation (5.2)
are zero, hence $\varphi$ sees $v$ with arbitrary value $c$. Thus $v$ is
ambiguous for $\varphi$.

(ii) follows from Part (i). $\square$

In the next 3 examples we calculate the ambiguous spaces of several
pseudomultipliers.

######  Example 5.4.

We showed in Example 2.14 that the pseudomultiplier $\varphi(z)=1/z$ on
$H^{2}$ has no ambiguities, directly from the definition of an ambiguous
vector. Proposition 5.3 reinforces this statement. The ambiguous vectors for
$\varphi$ are the members of $E_{\varphi}^{\perp}\cap\ker X_{\varphi}^{*}$.
Recall that $E_{\varphi}=zH^{2}$, and so $E_{\varphi}^{\perp}$ is the space
$H^{2}\ominus zH^{2}$ of constant functions on $\mathbb{D}$. Consider any
$f\in\mathcal{A}_{\varphi}$: then, since $f\in E_{\varphi}^{\perp}$,
$f=c\mathbf{1}$ for some $c\in\mathbb{C}$, and since $f\in\ker
X_{\varphi}^{*}$, $X_{\varphi}^{*}f=0$. Thus, for every $g\in H^{2}$,

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\left\langle
X_{\varphi}^{*}c\mathbf{1},zg\right\rangle$ $\displaystyle=$
$\displaystyle\left\langle
c\mathbf{1},X_{\varphi}zg\right\rangle=c\left\langle\mathbf{1},g\right\rangle$

and therefore $c=0$. Hence $\mathcal{A}_{\varphi}=\\{0\\}$. $\square$

######  Example 5.5.

Consider the pseudomultiplier $\chi$ in Example 2.5 ($\chi(t)=\sqrt{t}$ on
$W^{1,2}[0,1])$. Recall that $E_{\chi}=k_{0}^{\perp}$. Let us show that
$\mathcal{A}_{\chi}=\mathbb{C}k_{0}$. By Proposition 5.3, the ambiguous
vectors for $\chi$ are the members of $E_{\chi}^{\perp}\cap\ker X_{\chi}^{*}$.
We need to show that every vector $v=ck_{0}$ belongs to $\ker X_{\chi}^{*}$.
Note that, for every $f\in E_{\chi}=k_{0}^{\perp}$,

$\displaystyle\left\langle X_{\chi}^{*}ck_{0},f\right\rangle=\left\langle
ck_{0},X_{\chi}f\right\rangle=c\overline{(\chi f)(0)}=0.$

Therefore the ambiguous space
$\mathcal{A}_{\chi}=\mathcal{S}_{\chi}=E_{\chi}^{\perp}=\mathbb{C}k_{0}$.
$\square$

######  Example 5.6.

Let $\mathcal{H}$ be the Hilbert space of functions $f$ on $[0,1]$ such that
both $f$ and $f^{\prime}$ are absolutely continuous on $[0,1]$ and such that
the weak derivative $f^{\prime\prime}$ is in $L^{2}$, equipped with the inner
product

$\left\langle
f,g\right\rangle_{\mathcal{H}}=f(0)g\overline{(0)}+f^{\prime}(0)g^{\prime}\overline{(0)}+\int^{1}_{0}f^{\prime\prime}(t)\overline{g^{\prime\prime}(t)}dt.$

If $\chi(t)=\sqrt{t}$, then we claim that

$E_{\chi}=\\{f\in\mathcal{H}:f(0)=f^{\prime}(0)=0\\}.$

Also $E^{\perp}_{\chi}$ is a 2 dimensional subspace consisting of ambiguous
vectors.

###### Proof.

To prove that $E_{\chi}=\\{f:f(0)=f^{\prime}(0)=0\\}$, consider any function
$f\in\mathcal{H}$ such that $f(0)=f^{\prime}(0)=0$. We have to show that $\chi
f$ and $(\chi f)^{\prime}$ are absolutely continuous on $[0,1]$ and $(\chi
f)^{\prime\prime}$ is in $L^{2}$. It follows from Example 2.5 that the
function

$(\chi f)^{\prime}(t)=\frac{1}{2\sqrt{t}}f(t)+\sqrt{t}f^{\prime}(t)$

is in $L^{2}(0,1)\subset L^{1}(0,1)$, and so $\chi f$ is absolutely continuous
on $[0,1]$.

Now let us show that $(\chi f)^{\prime\prime}\in L^{2}(0,1)$. We have, for
$t\in(0,1]$,

$(\chi
f)^{\prime\prime}(t)=-\tfrac{1}{4}t^{-\tfrac{3}{2}}f(t)+\frac{1}{\sqrt{t}}f^{\prime}(t)+\sqrt{t}f^{\prime\prime}(t).$
(5.8)

The third term in the sum on the right hand side of equation (5.8) is in
$L^{2}(0,1)$, because $f^{\prime\prime}$ is in $L^{2}(0,1)$. The second term
in the sum of equation (5.8) is $\frac{1}{\sqrt{t}}f^{\prime}(t)$. Since
$f^{\prime}(0)=0$, for $t\in(0,1)$,

$\displaystyle t^{-\tfrac{1}{2}}f^{\prime}(t)$ $\displaystyle=$ $\displaystyle
t^{-\tfrac{1}{2}}\int_{0}^{t}f^{\prime\prime}(u)\,du$ (5.9) $\displaystyle=$
$\displaystyle\sqrt{t}\left(t^{-1}\int_{0}^{t}f^{\prime\prime}(u)\,du\right).$

By assumption $f^{\prime\prime}$ is in $L^{2}(0,1)$, and so, by Hardy’s
inequality, $\left(t^{-1}\int_{0}^{t}f^{\prime\prime}(u)\,du\right)$ is in
$L^{2}(0,1)$. Thus $\frac{1}{\sqrt{t}}f^{\prime}(t)$ is in $L^{2}(0,1)$.

We claim that the first term on the right hand side of equation (5.8) is in
$L^{\infty}(0,1)$. Indeed, for $x\in(0,1)$,

$\displaystyle x^{-\tfrac{3}{2}}f(x)$ $\displaystyle=$ $\displaystyle
x^{-\tfrac{3}{2}}\int_{0}^{x}f^{\prime}(t)\,dt\qquad\qquad(\text{let }u=t/x)$
(5.10) $\displaystyle=$ $\displaystyle
x^{-\tfrac{3}{2}}\int_{0}^{1}f^{\prime}(xu)x\,du$ $\displaystyle=$
$\displaystyle x^{-\tfrac{1}{2}}\int_{0}^{1}f^{\prime}(xu)\,du$
$\displaystyle=$ $\displaystyle
x^{-\tfrac{1}{2}}\int_{0}^{1}\int_{0}^{xu}f^{\prime\prime}(t)\,dt\,du.$

By Hölder’s inequality,

$\left|\int_{0}^{xu}f^{\prime\prime}(\xi)\
d\xi\right|\leq|xu|^{\tfrac{1}{2}}\left\\{\int_{0}^{xu}|f^{\prime\prime}(\xi)|^{2}\
d\xi\right\\}^{\tfrac{1}{2}}\leq|xu|^{\tfrac{1}{2}}\|f^{\prime\prime}\|_{L^{2}}.$
(5.11)

Therefore

$\displaystyle|x^{-\tfrac{3}{2}}f(x)|$ $\displaystyle=$ $\displaystyle
x^{-\tfrac{1}{2}}\left|\int_{0}^{1}\int_{0}^{xu}f^{\prime\prime}(t)\,dt\,du\right|$
(5.12) $\displaystyle\leq$ $\displaystyle
x^{-\tfrac{1}{2}}\int_{0}^{1}|xu|^{\tfrac{1}{2}}\|f^{\prime\prime}\|_{L^{2}}\,du$
$\displaystyle=$
$\displaystyle\|f^{\prime\prime}\|_{L^{2}}\int_{0}^{1}u^{{\tfrac{1}{2}}}\,du=\tfrac{2}{3}\|f^{\prime\prime}\|_{L^{2}},$

so that $x^{-\tfrac{3}{2}}f(x)$ is bounded on $[0,1]$. Hence the weak
derivative $(\chi f)^{\prime\prime}\in L^{2}(0,1)$, and consequently $(\chi
f)^{\prime\prime}\in L^{1}(0,1)$. Therefore

$(\chi f)^{\prime}(x)-(\chi f)^{\prime}(0)=\int_{0}^{x}(\chi
f)^{\prime\prime}(t)\ dt.$

Note that, by equation (2.5) and the fact that $f(0)=0$,

$(\chi
f)^{\prime}(t)=\frac{1}{2\sqrt{t}}f(t)+\sqrt{t}f^{\prime}(t)=\frac{1}{2}\sqrt{t}\left(t^{-1}\int_{0}^{t}f^{\prime}(u)\,du\right)+\sqrt{t}f^{\prime}(t).$

Since $f^{\prime}$ is in $L^{2}(0,1)$, by Hardy’s inequality,
$\left(t^{-1}\int_{0}^{t}f^{\prime}(u)\,du\right)$ is in $L^{2}(0,1)$, and so,
because $f^{\prime}(0)=0$, $(\chi f)^{\prime}(0)=0$. Hence $(\chi f)^{\prime}$
is absolutely continuous on $[0,1]$.

Therefore, for $f\in\mathcal{H}$ such that $f(0)=f^{\prime}(0)=0$, $\chi
f\in\mathcal{H}$. Conversely, if $f\in\mathcal{H}$ is such that the condition
$f(0)=f^{\prime}(0)=0$ is not satisfied, that is $f(0)\neq 0$ or
$f^{\prime}(0)\neq 0$ then $(\chi f)^{\prime\prime}\notin L^{2}[0,1]$ and so
$\chi f\notin\mathcal{H}$. Thus, $\chi$ is a pseudomultiplier of $\mathcal{H}$
and $E_{\chi}=\\{f\in\mathcal{H}:f(0)=f^{\prime}(0)=0\\}$. For
$f\in\mathcal{H}$, we have $|f(0)|\leq\|f\|_{\mathcal{H}}$ and
$|f^{\prime}(0)|\leq\|f\|_{\mathcal{H}}$, and so there exist
$k_{0},s_{0}\in\mathcal{H}$ such that, for all $f\in\mathcal{H}$,
$\left\langle f,k_{0}\right\rangle=f(0)$ and $\left\langle
f,s_{0}\right\rangle=f^{\prime}(0)$. Thus
$E_{\chi}^{\perp}=\operatorname{span}\\{k_{0},s_{0}\\}$,

Let us show that $\mathcal{A}_{\chi}=\mathbb{C}k_{0}+\mathbb{C}s_{0}$. By
Proposition 5.3, the ambiguous vectors for $\chi$ are the members of
$E_{\chi}^{\perp}\cap\ker X_{\chi}^{*}$. We need to show that every vector
$v=c_{1}k_{0}+c_{2}s_{0}$ belongs to $\ker X_{\chi}^{*}$. Note that, for every
$f\in E_{\chi}=\\{f\in\mathcal{H}:f(0)=f^{\prime}(0)=0\\}$, since $(\chi
f)(0)=(\chi f)^{\prime}(0)=0$,

$\displaystyle\left\langle
X_{\chi}^{*}(c_{1}k_{0}+c_{2}s_{0}),f\right\rangle=\left\langle
c_{1}k_{0}+c_{2}s_{0},X_{\chi}f\right\rangle=c_{1}\overline{(\chi
f)(0)}+c_{2}\overline{(\chi f)^{\prime}(0)}=0.$

Therefore the ambiguous space
$\mathcal{A}_{\chi}=\mathcal{S}_{\chi}=E_{\chi}^{\perp}=\operatorname{span}\\{k_{0},s_{0}\\}$.
$\square$

##  6\. Definable vectors and polar vectors

In this section we shall study definable vectors, polar vectors, and, more
generally, the polar space of a pseudomultiplier $\varphi$. By way of a
preliminary observation relevant to the notion of a definable vector, we note
that, if $\varphi$ sees $v$ then $\varphi$ is constant on the coset
$v+\mathcal{A}_{\varphi}$.

######  Lemma 6.1.

Let $\varphi$ be a pseudomultiplier, let $v\in\mathcal{H}$, and let
$a\in\mathcal{A}_{\varphi}$. If $\varphi$ sees $v$ with value $c$, then
$\varphi$ sees $v+a$ with value $c$. In particular, if $\varphi$ sees $v$,
then $\varphi$ sees $v+a$.

###### Proof.

Suppose $\varphi$ sees $v$ with value $c$. Thus, by Lemma 5.1, there exists
$u\in E^{\perp}_{\varphi}$ such that

$X^{*}_{\varphi}v=\overline{c}v+u.$

Noting that $X^{*}_{\varphi}a=0$, we see that

$\displaystyle X^{*}_{\varphi}(v+a)$ $\displaystyle=$
$\displaystyle\overline{c}v+u$ $\displaystyle=$
$\displaystyle\overline{c}(v+a)+u-\overline{c}a.$

Since $\overline{c}a\in E^{\perp}_{\varphi}$, so also $u-\overline{c}a\in
E^{\perp}_{\varphi}$. Hence by Lemma 5.1, $\varphi$ sees $v+a$ with value $c$.
$\square$

If $\varphi$ is a pseudomultiplier of $\mathcal{H}$ then $\varphi$ naturally
extends to have values on its visible vectors. These values are arbitrary on
the ambiguous vectors but unique on the unambiguous visible vectors. We now
know from Lemma 6.1, that $\varphi$ is constant on each coset
$v+\mathcal{A}_{\varphi}$. Since, for any visible vector $v$, the coset
$v+\mathcal{A}_{\varphi}$ contains a unique visible vector that is orthogonal
to $\mathcal{A}_{\varphi}$, to wit $P_{{\mathcal{A}_{\varphi}^{\perp}}}v$, the
following definition is reasonable.

######  Definition 6.2.

Let $\varphi$ be a pseudomultiplier. We say a vector $d\in\mathcal{H}$ is
definable for $\varphi$ if $d\not=0,\varphi$ sees $d$, and
$d\perp\mathcal{A}_{\varphi}$. We let $\mathbf{D}_{\varphi}$ denote the set of
vectors that are definable for $\varphi$. If $d$ is definable for $\varphi$
then there exists a unique $c\in\mathbb{C}$ such that $\varphi$ sees $d$ with
value $c$. We refer to this $c$ as the value of $\varphi$ at $d$ and denote it
by $\varphi(d)$.

We remark that the exclusion of the vector $0$ from being definable has the
following happy consequences. No vector is both definable and ambiguous, and,
by Lemma 6.1, every vector in $\mathbf{D}_{\varphi}+\mathcal{A}_{\varphi}$ is
an unambiguously seen vector. It is also clear that if $v$ is an unambiguously
seen vector, then $v$ has a unique representation $v=d+a$ with
$d\in\mathbf{D}_{\varphi}$ and $a\in\mathcal{A}_{\varphi}$.

######  Remark 6.3.

We proved in Example 2.16 that, for the pseudomultiplier $\varphi(z)=1/z$ on
$H^{2}$, the set of definable vectors $\mathbf{D}_{\varphi}$ comprises the
non-zero scalar multiples of the kernels $k_{\lambda}$, for
$\lambda\in\mathbb{D}\setminus\\{0\\}$. The vector $k_{0}$ is a limit of the
definable vectors $k_{(1/n)}$ for $\varphi$

$k_{0}=\lim\limits_{n\to\infty}k_{(1/n)}.$

However, $k_{0}$ is not itself definable because, for all $n\geq 1$,
$\varphi(k_{1/n})=n$, and so

$\lim_{n\to\infty}\varphi(k_{1/n})=\infty.$

Thus $k_{0}$ is a polar vector for $\varphi$.

$\mathcal{A}_{\varphi}$ is both a subspace and a topologically closed set in
$\mathcal{H}$. Not so $\mathbf{D}_{\varphi}$. Quite apart from the fact that
$0$ is not in $\mathbf{D}_{\varphi}$, in the example above
$\mathbf{D}_{\varphi}$ is very far from being a subspace but not far from
being a closed set in $\mathcal{H}$. For example, in Example 2.16 one need
only adjoin $\mathbb{C}k_{0}$ to $\mathbf{D}_{\varphi}$ to obtain a closed
set. $\square$

We shall repeatedly have a use for the following minor technical observation.

######  Lemma 6.4.

If $d$ is a definable vector of a pseudomultiplier $\varphi$ then
$P_{E_{\varphi}}d\neq 0$.

###### Proof.

Suppose $P_{E_{\varphi}}d=0$, and so $d\in E_{\varphi}^{\perp}$. By Lemma 5.1,
since $d$ is visible for $\varphi$ with value $\varphi(d)$,

$X_{\varphi}^{*}d=\overline{\varphi(d)}P_{E_{\varphi}}d=0,$

and thus $d\in E_{\varphi}^{\perp}\cap\ker X_{\varphi}^{*}$. Hence, by
Proposition 5.3, $d\in\mathcal{A}_{\varphi}$. However, by the definition of
definable vectors, $d\neq 0$ and $d\perp\mathcal{A}_{\varphi}$, which is a
contradiction. Hence $P_{E_{\varphi}}d\neq 0$. $\square$

######  Proposition 6.5.

Let $\varphi$ be a pseudomultiplier. Then $\varphi$ is a continuous function
on $\mathbf{D}_{\varphi}$ with respect to the metric of $\mathcal{H}$.

###### Proof.

Since $\mathbf{D}_{\varphi}$, with the topology inherited from $\mathcal{H}$,
is a metric space, it suffices to prove sequential continuity. Let $d_{n}\to
d$ in $\mathbf{D}_{\varphi}$. By Lemma 5.1,

$\displaystyle X^{*}_{\varphi}d_{n}$ $\displaystyle=$
$\displaystyle\overline{\varphi(d_{n})}\ P_{E_{\varphi}}d_{n},$ $\displaystyle
X^{*}_{\varphi}d$ $\displaystyle=$ $\displaystyle\overline{\varphi(d)}\
P_{E_{\varphi}}d.$

By Lemma 6.4, $P_{E_{\varphi}}d\neq 0$. On letting $n\to\infty$ we therefore
find that $\varphi(d_{n})\to\varphi(d)$. $\square$

Recall that we introduced the polar vectors of a pseudomultiplier in
Definition 2.17: for a pseudomultiplier $\varphi$ of $\mathcal{H}$ and for
$p\in\mathcal{H}$, we say that $p$ is a polar vector for $\varphi$ if
$p\not=0$ and there exists a sequence
$\\{d_{n}\\}\subseteq\mathbf{D}_{\varphi}$ such that $d_{n}\to p$ (with
respect to $\|\cdot\|_{\mathcal{H}}$) and $\varphi(d_{n})\to\infty$ as
$n\to\infty$.  We define the polar space $\mathcal{P}_{\varphi}$ of $\varphi$
to be the set of vectors $v\in\mathcal{H}$ such that either $v$ is a polar
vector of $\varphi$ or $v=0$.

######  Example 6.6.

Let $\mathcal{H}=H^{2}$, the classical Hardy space on $\mathbb{D}$, let $n\geq
2$, and define $\varphi$ on $\mathbb{D}$ by

$\varphi(\lambda)=\left\\{\begin{array}[]{cc}\frac{1}{\lambda^{n}}&\qquad\text{
if }\lambda\not=0\\\ \\\ 1&\qquad\text{ if }\lambda=0.\end{array}\right.$

Here

$E_{\varphi}=\\{f\in
H^{2}:f(0)=f^{\prime}(0)=\dots=f^{(n)}(0)=0\\}=z^{n+1}H^{2}.$

Let us describe the space of ambiguous vectors $\mathcal{A}_{\varphi}$ for
$\varphi$. By Proposition 5.3,

$\mathcal{A}_{\varphi}=E_{\varphi}^{\perp}\cap\ker X_{\varphi}^{*}.$

The operator $X_{\varphi}:z^{n+1}H^{2}\to H^{2}$ is given by
$X_{\varphi}=(S^{*})^{n}P_{z^{n+1}H^{2}}^{*}$ where $S$ denotes the forward
shift operator on $H^{2}$ and $P_{z^{n+1}H^{2}}:H^{2}\to z^{n+1}H^{2}$ is the
orthogonal projection operator (so that $P_{z^{n+1}H^{2}}^{*}$ is the
injection operator $z^{n+1}H^{2}\to H^{2}$). Thus
$X_{\varphi}^{*}=P_{z^{n+1}H^{2}}S^{n}$ and so, for $f\in H^{2}$,

$\displaystyle X_{\varphi}^{*}f=0$ $\displaystyle\Leftrightarrow$
$\displaystyle S^{n}f\in H^{2}\ominus z^{n+1}H^{2}$
$\displaystyle\Leftrightarrow$ $\displaystyle z^{n}f\in$span{1,z, …, z^n}
$\displaystyle\Leftrightarrow$ $\displaystyle f\text{ is constant.}$

Hence, the space of ambiguous vectors for $\varphi$ is

$\mathcal{A}_{\varphi}=E_{\varphi}^{\perp}\cap\ker
X_{\varphi}^{*}=(H^{2}\ominus
z^{n+1}H^{2})\cap\mathbb{C}\mathbf{1}=\mathbb{C}\mathbf{1}.$

Let us describe the set of definable vectors $\mathbf{D}_{\varphi}$ of
$\varphi$. By Lemma 5.1, for $f\in H^{2}$, $\varphi$ sees $f$ with value $c$
if and only if there exists $u\in E^{\perp}_{\varphi}$ such that

$X^{*}_{\varphi}f=\overline{c}f+u,$ (6.2)

equivalently, if and only if there exist
$a_{0},a_{1},\dots,a_{n}\in\mathbb{C}$ such that

$X_{\varphi}^{*}f=\bar{c}f+a_{0}+a_{1}z+\dots+a_{n}z^{n},$

that is, if and only if there exist $a_{0},a_{1},\dots,a_{n}\in\mathbb{C}$
such that

$P_{z^{n+1}H^{2}}z^{n}f-\bar{c}f=a_{0}+a_{1}z+\dots+a_{n}z^{n}.$ (6.3)

Now $P_{z^{n+1}H^{2}}z^{n}f=z^{n}(f-f(0))$, and so equation (6.3) is
equivalent to

$z^{n}(f-f(0))-\bar{c}f=a_{0}+a_{1}z+\dots+a_{n}z^{n}.$ (6.4)

Substitution of $z=0$ shows that equation (6.4) implies that
$a_{0}=-\bar{c}f(0)$, and so equation (6.4) can be written

$(z^{n}-\bar{c})(f-f(0))=a_{1}z+\dots+a_{n}z^{n}.$

Thus $\varphi$ sees $f$ with value $c$ if and only if there exist
$a_{1},a_{2},\dots,a_{n}\in\mathbb{C}$ such that

$f=f(0)+\frac{a_{1}z+\dots+a_{n}z^{n}}{z^{n}-\bar{c}}\in H^{2}.$

The rational function on the right hand side is in $H^{2}$ if and only if
$|c|>1$, and so the visible vectors for $\varphi$ consist of the rational
functions of the form

$f(z)=a_{0}+\frac{a_{1}z+\dots+a_{n}z^{n}}{z^{n}-\bar{c}}\text{ where
}|c|>1\text{ and }a_{0},a_{1},\dots,a_{n}\in\mathbb{C},$

or alternatively, if we define $b_{j}=-\bar{c}^{-1}a_{j}$ for $j=1,2,\dots,n$,
of the functions

$f(z)=a_{0}+\frac{b_{1}z+b_{2}z^{2}+\dots+b_{n}z^{n}}{1-\bar{c}^{-1}z^{n}}\text{
where }|c|>1\text{ and }a_{0},b_{1},b_{2},\dots,b_{n}\in\mathbb{C}.$ (6.5)

We showed above that $\mathcal{A}_{\varphi}$ consists of the constant
functions, and so the visible function $f$ in equation (6.5) is definable if
and only if $f\neq 0$ but $f(0)=0$. Thus

$\mathbf{D}_{\varphi}=\left\\{f(z)=\frac{b_{1}z+b_{2}z^{2}+\dots+b_{n}z^{n}}{1-\bar{c}^{-1}z^{n}}:|c|>1\text{
and }b_{1},b_{2},\dots,b_{n}\in\mathbb{C}\text{ are not all zero}\right\\}.$
(6.6)

Now we can describe the polar space $\mathcal{P}_{\varphi}$ for $\varphi$. For
every non-zero polar vector $p$ there exists a convergent sequence of
definable vectors

$\left\\{d_{k}=\frac{b_{1k}z+b_{2k}z^{2}+\dots+b_{nk}z^{n}}{1-\bar{c_{k}}^{-1}z^{n}}\right\\}_{k=1}^{\infty}\text{
in }\mathbf{D}_{\varphi},$

where $b_{1k},b_{2k},\dots,b_{nk}\in\mathbb{C}$, not all zero, are such that
$d_{k}\to p$ and $\varphi(d_{k})=c_{k}\to\infty$ as $k\to\infty$. The non-zero
limits of such convergent sequences $\\{d_{k}\\}_{k=1}^{\infty}$ in $H^{2}$
coincide with $\operatorname{span}\\{z,z^{2},\dots,z^{n}\\}\setminus\\{0\\}$,
so that $\mathcal{P}_{\varphi}=\operatorname{span}\\{z,z^{2},\dots,z^{n}\\}$.
We may observe that

$\mathcal{A}_{\varphi}\oplus\mathcal{P}_{\varphi}=\mathbb{C}\mathbf{1}\oplus\operatorname{span}\\{z,z^{2},\dots,z^{n}\\}=H^{2}\ominus
z^{n+1}H^{2}=E_{\varphi}^{\perp}.$

$\square$

######  Remark 6.7.

The combination of Examples 6.6 and 5.4 (with $n=1$) illustrates an
interesting fact: by undefining the pseudomultiplier $\varphi$ of Example 6.6
at the point $0$ we obtain a pseudomultiplier with a smaller singular space –
see Example 5.4. The polar space is unchanged, while the ambiguous space is
reduced from $\mathbb{C}\mathbf{1}$ in Example 6.6 to $\\{0\\}$ in Example
5.4.

The following proposition shows how the notion of a pseudopole of a
pseudomultiplier, as introduced in [5] (see Remark 2.18 above), relates to
polar vectors.

######  Proposition 6.8.

If $\alpha$ is a pseudopole of a pseudomultiplier $\varphi$ and $\alpha$ is in
the closure of $D_{\varphi}$ in $\Omega$ then
$P_{\mathcal{A}_{\varphi}^{\perp}}k_{\alpha}$ is a polar vector of $\varphi$.

###### Proof.

By assumption, $\alpha$ is in the closure of $D_{\varphi}$ in $\Omega$, and so
we can choose a sequence $(\alpha_{n})$ in $D_{\varphi}$ such that
$\alpha_{n}\to\alpha$ as $n\to\infty$. Since $\alpha$ is a pseudopole of
$\varphi$ there exists $h\in E_{\varphi}$ and $g\in\mathcal{H}$ such that
$g=X_{\varphi}h$, $h(\alpha)=0$ and $g(\alpha)\neq 0$. For $n\geq 1$ let

$d_{n}=P_{\mathcal{A}_{\varphi}^{\perp}}k_{\alpha_{n}}=k_{\alpha_{n}}-P_{\mathcal{A}_{\varphi}}k_{\alpha_{n}}.$

Since $\varphi$ sees $k_{\alpha_{n}}$ with value $\varphi(\alpha_{n})$, it
follows from Lemma 6.1 that $\varphi$ sees $d_{n}$ with value
$\varphi(\alpha_{n})$. Since $d_{n}\perp\mathcal{A}_{\varphi}$, $d_{n}$ is a
definable vector for $\varphi$ and $\varphi(d_{n})=\varphi(\alpha_{n})$.
Moreover

$0\neq g(\alpha)=\left\langle g,k_{\alpha}\right\rangle=\lim_{n}\left\langle
g,k_{\alpha_{n}}\right\rangle,$

while, for all $\lambda\in D_{\varphi}$,

$\left\langle
g,k_{\lambda}\right\rangle=g(\lambda)=\varphi(\lambda)h(\lambda).$

Hence

$\lim_{n}\left\langle
g,k_{\alpha_{n}}\right\rangle=\lim_{n}\varphi(\alpha_{n})h(\alpha_{n})=g(\alpha)\neq
0.$

Since also $h(\alpha_{n})\to h(\alpha)=0$, we have
$\varphi(\alpha_{n})\to\infty$ and therefore $\varphi(d_{n})\to\infty$ as
$n\to\infty$. Since the sequence of definable vectors
$d_{n}=P_{\mathcal{A}_{\varphi}^{\perp}}k_{\alpha_{n}}$ tends to
$P_{\mathcal{A}_{\varphi}^{\perp}}k_{\alpha}$ as $n\to\infty$ it follows that
$P_{\mathcal{A}_{\varphi}^{\perp}}k_{\alpha}$ is a polar vector of $\varphi$.
$\square$

The nature of polar vectors is much more rigid than one might at first expect.
This is brought out in the following proposition, which gives a clean
relationship between $\mathbf{D}_{\varphi}$ and the set
$\mathcal{P}_{\varphi}$ of polar vectors of $\varphi$.

######  Proposition 6.9.

Let $\varphi$ be a pseudomultiplier and let $p\in\mathcal{H}$ with $p\not=0$.
The following are equivalent.

  1. (i)

$p$ is a polar vector for $\varphi$.

  2. (ii)

$p\in\mathbf{D}_{\varphi}^{-}$ and for every neighborhood $U$ of $p,\varphi$
is unbounded on $U\cap\mathbf{D}_{\varphi}$.

  3. (iii)

$p\in\mathbf{D}_{\varphi}^{-}\backslash\mathbf{D}_{\varphi}$.

  4. (iv)

$p\in\mathbf{D}_{\varphi}^{-}$ and
$\lim\limits_{\stackrel{{\scriptstyle\textstyle{d\to
p}}}{{d\in\mathbf{D}_{\varphi}}}}\ \varphi(d)=\infty$.

###### Proof.

It is clear that (iv) $\Rightarrow$ (i) $\Rightarrow$ (ii). Suppose that (ii)
holds but (iii) is false, so that $p\in\mathbf{D}_{\varphi}$. By Lemma 6.4,
$P_{E_{\varphi}}d\neq 0$. We may therefore assume $\|P_{E_{\varphi}}p\|=1$.
Let $U$ be a neighborhood of $p$ in $\mathcal{H}$ such that, for all $v\in U$,

$\|(X^{*}_{\varphi}-\overline{\varphi(p)}\ P_{E_{\varphi}})v\|<1$ (6.7)

and

$\|P_{E_{\varphi}}v\|>\frac{1}{2}.$

Consider $v\in\mathbf{D}_{\varphi}\cap U$. Since $v$ is definable,

$X^{*}_{\varphi}v=\overline{\varphi(v)}\ P_{E_{\varphi}}v.$

Substituting in (6.7) we have

$\|(\overline{\varphi(v)}-\overline{\varphi(p)})\ P_{E_{\varphi}}v\|<1$

and so

$|\varphi(v)-\varphi(p)|<2.$

Thus $\varphi$ is bounded on $U\cap\mathbf{D}_{\varphi}$, contradicting (ii).
Thus (ii) $\Rightarrow$ (iii).

Now suppose (iii) holds but (iv) is false:
$p\in\mathbf{D}_{\varphi}^{-}\smallsetminus\mathbf{D}_{\varphi}$ but
$\varphi(d)\not\rightarrow\infty$ as $d\to p$ in $\mathbf{D}_{\varphi}$. Then
there exists a sequence $(d_{n})$ in $\mathbf{D}_{\varphi}$ and a positive
constant $C$ such that $d_{n}\to p$ and $|\varphi(d_{n})|\leq C$ for all $n$.
By Lemma 5.1, for all $n\in\mathbb{N}$,

$X^{*}_{\varphi}d_{n}=\varphi\overline{(d_{n})}\ P_{E_{\varphi}}d_{n}.$

Passing to a subsequence if necessary, we can assume $\varphi(d_{n})\to
z\in\mathbb{C}$. Taking limits in the foregoing equation we find

$X^{*}_{\varphi}p=\overline{z}P_{E_{\varphi}}p.$

Thus $\varphi$ sees $p$ with value $z$. Since each
$d_{n}\perp\mathcal{A}_{\varphi}$, we have also $p\perp\mathcal{A}_{\varphi}$.
By assumption $p\not=0$, and so $p\in\mathbf{D}_{\varphi}$, a contradiction.
Thus (iii) $\Rightarrow$ (iv). $\square$

We continue this section with a fact about $\varphi$ as a function on
$\mathbf{D}_{\varphi}$ (a corollary of Proposition 6.9).

We say that a pseudomultiplier $\varphi$ is locally bounded on
$\mathbf{D}_{\varphi}^{-}$ if, for every point $d\in\mathbf{D}_{\varphi}^{-}$
there is a neighbourhood $U$ of $d$ in $\mathcal{H}$ such that $\varphi$ is
bounded on $U\cap\mathbf{D}_{\varphi}$.

######  Theorem 6.10.

Let $\varphi$ be a pseudomultiplier. Then $\varphi$ is locally bounded on
$\mathbf{D}_{\varphi}^{-}$ if and only if $\varphi$ has no polar vectors.

###### Proof.

Suppose $\varphi$ has a polar vector $p$. By Proposition 6.9,
(i)$\Leftrightarrow$(ii), for every neighbourhood $U$ of $p$ in $\mathcal{H}$,
$\varphi$ is unbounded on $U\cap\mathbf{D}_{\varphi}$. Thus $\varphi$ is not
locally bounded on $\mathbf{D}_{\varphi}^{-}$.

Conversely, suppose $\varphi$ has no polar vectors. By Proposition 6.9,
(i)$\Leftrightarrow$(ii), for every point $d\in\mathbf{D}_{\varphi}^{-}$ there
is a neighbourhood $U$ of $d$ in $\mathcal{H}$ such that $\varphi$ is bounded
on $U\cap\mathbf{D}_{\varphi}$, that is, $\varphi$ is locally bounded on
$\mathbf{D}_{\varphi}^{-}$. $\square$

The definable vectors provide an alternative viewpoint on the polar space.

######  Theorem 6.11.

Let $\varphi$ be a pseudomultiplier of a Hilbert function space $\mathcal{H}$
on a set $\Omega$. Suppose $\dim\mathcal{H}=\infty$. Then

(i) $\mathbf{D}_{\varphi}$ is non-empty.

(ii)
$\mathcal{P}_{\varphi}=\mathbf{D}_{\varphi}^{-}\smallsetminus\mathbf{D}_{\varphi}$.

###### Proof.

(i). Let us show that $\mathbf{D}_{\varphi}=\emptyset$ implies that
$\mathcal{H}$ is finite-dimensional.

For any point $\alpha\in D_{\varphi}$, let
$d_{\alpha}\stackrel{{\scriptstyle\rm
def}}{{=}}P_{\mathcal{A}^{\perp}_{\varphi}}k_{\alpha}$. Let us show that
$d_{\alpha}$ is definable if and only if $\alpha$ is not ambiguous. Suppose
that $\alpha$ is ambiguous. Then by Theorem 4.14,
$k_{\alpha}\in\mathcal{A}_{\varphi}$, and so $d_{\alpha}=0$, and therefore
$d_{\alpha}$ is not definable. Suppose now that $\alpha$ is not ambiguous.
Then $\varphi$ sees $k_{\alpha}$ with the unique value $\varphi(\alpha)$. By
definition $d_{\alpha}\perp\mathcal{A}_{\varphi}$. By Lemma 6.1, $\varphi$
sees $d_{\alpha}$ with the unique value $\varphi(\alpha)$. Thus $d_{\alpha}$
is definable.

By assumption, $\mathbf{D}_{\varphi}=\emptyset$, and so, for every $\alpha\in
D_{\varphi}$, $d_{\alpha}$ is not definable, that is, every $\alpha\in
D_{\varphi}$ is ambiguous. Hence $\mathcal{A}_{\varphi}$ contains
$\\{k_{\alpha}:\alpha\in D_{\varphi}\\}$, and so also its closed linear span,
which is $\mathcal{H}$. By Proposition 5.3,
$\mathcal{A}_{\varphi}=E_{\varphi}^{\perp}\cap\mathrm{ker}X_{\varphi}^{*}$,
which is a finite-dimensional subspace of $\mathcal{H}$. This implies that
$\mathcal{H}$ is finite-dimensional. Therefore if $\mathcal{H}$ is infinite-
dimensional, then $\mathbf{D}_{\varphi}$ is non-empty.

(ii) Since $\mathcal{H}$ is infinite-dimensional, by (i),
$\mathbf{D}_{\varphi}$ is non-empty: pick $d\in\mathbf{D}_{\varphi}$. By
Corollary 5.2, $n^{-1}d\in\mathbf{D}_{\varphi}$ for $n\in\mathbb{N}$, and so
$0\in\overline{\mathbf{D}}_{\varphi}$. By definition,
$0\not\in\mathbf{D}_{\varphi}$ and so
$0\in\overline{\mathbf{D}_{\varphi}}\smallsetminus\mathbf{D}_{\varphi}$. The
result is now immediate from the equivalence of (i) and (iii) in Proposition
6.9. $\square$

##  7\. The singular space

Pseudomultipliers fail to act as operators on all of $\mathcal{H}$ for one of
two reasons. Either they have an ambiguous point (such as $\sqrt{t}$ acting on
Sobolev space) or they have a polar vector (such as $\frac{1}{z}$ acting on
$H^{2}$). In this section we prove a theorem to the effect that that every
singularity of a pseudomultiplier can be written in a unique way as the sum of
a polar vector and an ambiguous vector. As a corollary of our basic theorem
one obtains the interesting fact that if one adjoins the zero vector to the
set of polar vectors of a pseudomultiplier then one obtains a finite-
dimensional subspace of $\mathcal{H}$. Furthermore, this subspace is precisely
the set of vectors that can be approximated by definable vectors but that are
not themselves definable.

######  Definition 7.1.

Let $\varphi$ be a pseudomultiplier. We define the order of $\varphi$ to be
$\dim E^{\perp}_{\varphi}$. If the order of $\varphi$ is $m$, then we say that
$\varphi$ is an $m$-pseudomultiplier.

######  Theorem 7.2.

Let $\varphi$ be a pseudomultiplier. $\mathcal{P}_{\varphi}$ is a closed
subspace of $\mathcal{H}$ and

$\mathcal{S}_{\varphi}=\mathcal{P}_{\varphi}\oplus\mathcal{A}_{\varphi}.$

###### Proof.

First recall that, by Proposition 5.3,
$\mathcal{A}_{\varphi}=E^{\perp}_{\varphi}\cap\ker X^{*}_{\varphi}$ is a
closed subspace of $\mathcal{S}_{\varphi}=E^{\perp}_{\varphi}$. Hence we need
to show that

$\mathcal{P}_{\varphi}=\mathcal{S}_{\varphi}\ominus\mathcal{A}_{\varphi}.$
(7.1)

We first show that
$\mathcal{P}_{\varphi}\subseteq\mathcal{S}_{\varphi}\ominus\mathcal{A}_{\varphi}$.
Consider $p\in\mathcal{P}_{\varphi}$. We can suppose $p\not=0$, so that $p$ is
a polar vector for $\varphi$. Choose
$\\{d_{n}\\}\subseteq\mathbf{D}_{\varphi}$ with $d_{n}\to p$ and
$\varphi(d_{n})\to\infty$. We claim that $p\in\mathcal{S}_{\varphi}$. By Lemma
5.1,

$X^{*}_{\varphi}d_{n}=\overline{\varphi(d_{n})}\big{(}d_{n}-P_{\mathcal{S}_{\varphi}}d_{n}\big{)}.$
(7.2)

Since $X^{*}_{\varphi}d_{n}\to X^{*}_{\varphi}p$ and $\varphi(d_{n})\to\infty$
we conclude from equation (7.2) that $d_{n}-P_{\mathcal{S}_{\varphi}}d_{n}\to
0$. Hence $p-P_{\mathcal{S}_{\varphi}}p=0$, that is,
$p\in\mathcal{S}_{\varphi}$. Next, since $d_{n}\perp\mathcal{A}_{\varphi}$, so
also $p=\lim d_{n}\perp\mathcal{A}_{\varphi}$. Thus
$p\in\mathcal{S}_{\varphi}\cap\mathcal{A}_{\varphi}^{\perp}=\mathcal{S}_{\varphi}\ominus\mathcal{A}_{\varphi}$,
and we have proved that
$\mathcal{P}_{\varphi}\subseteq\mathcal{S}_{\varphi}\ominus\mathcal{A}_{\varphi}$.

We now show that
$\mathcal{S}_{\varphi}\ominus\mathcal{A}_{\varphi}\subseteq\mathcal{P}_{\varphi}$.
Consider $v\in\mathcal{S}_{\varphi}\ominus\mathcal{A}_{\varphi}$. If $v=0$,
then obviously $v\in\mathcal{P}_{\varphi}$. Assume that $v\not=0$. We claim
that $v$ is a polar vector for $\varphi$. Choose a sequence
$\\{c_{n}\\}\subseteq\mathbb{C}$ with $c_{n}\to\infty$ and
$\overline{c}_{n}\not\in\sigma(X^{*}_{\varphi})$ for all $n$. Define
$v_{n}\in\mathcal{H}$ by the formula

$v_{n}=\overline{c}_{n}\left(\overline{c}_{n}I-X^{*}_{\varphi}\right)^{-1}v.$
(7.3)

We claim that $\varphi$ sees $v_{n}$ with value $c_{n}$. We have

$\displaystyle X^{*}_{\varphi}v_{n}$ $\displaystyle=$ $\displaystyle
X^{*}_{\varphi}\overline{c}_{n}\left(\overline{c}_{n}I-X^{*}_{\varphi}\right)^{-1}v$
(7.4) $\displaystyle=$
$\displaystyle\overline{c}_{n}\left(\left(X^{*}_{\varphi}-\overline{c}_{n}I\right)\left(\overline{c}_{n}I-X^{*}_{\varphi}\right)^{-1}v+\overline{c}_{n}\left(\overline{c}_{n}I-X^{*}_{\varphi}\right)^{-1}v\right)$
$\displaystyle=$ $\displaystyle\overline{c}_{n}v_{n}-\overline{c}_{n}v$

and $v\in\mathcal{S}_{\varphi}$. Hence by Lemma 5.1, $\varphi$ sees $v_{n}$
with value $c_{n}$. Now define $w_{n}$ by

$w_{n}=P_{\mathcal{A}_{\varphi}^{\perp}}v_{n}=v_{n}-P_{\mathcal{A}_{\varphi}}v_{n}.$

Since $\varphi$ sees $v_{n}$ with value $c_{n}$, it follows from Lemma 6.1
that $\varphi$ sees $w_{n}$ with value $c_{n}$, and so $w_{n}$ is a definable
vector for $\varphi$. Note that $w_{n}\perp\mathcal{A}_{\varphi}$ by
definition of $w_{n}$. Furthermore, from equation (7.3) we see that $v_{n}\to
v$ as $n\to\infty$. By assumption
$v\in\mathcal{S}_{\varphi}\ominus\mathcal{A}_{\varphi}$, and so
$v\perp\mathcal{A}_{\varphi}$. Thus $w_{n}\to
P_{\mathcal{A}_{\varphi}^{\perp}}v=v$, and for sufficiently large
$n,w_{n}\not=0$. Consequently, for sufficiently large
$n,w_{n}\in\mathbf{D}_{\varphi}$ and $\varphi(w_{n})=c_{n}$. Since $w_{n}\to
v$ and $c_{n}\to\infty$ we conclude that $v$ is a polar vector. Thus
$\mathcal{S}_{\varphi}\ominus\mathcal{A}_{\varphi}\subseteq\mathcal{P}_{\varphi}$.
Therefore $\mathcal{P}_{\varphi}$ is a closed subspace of $\mathcal{H}$ and

$\mathcal{S}_{\varphi}=\mathcal{P}_{\varphi}\oplus\mathcal{A}_{\varphi}.$

$\square$

Theorem 7.2 gives a concrete expression of the polar space in terms of the
singular and ambiguous spaces.

######  Example 7.3.

Let $\mathcal{H}=H^{2}$, the classical Hardy space on $\mathbb{D}$. Consider
the function $\varphi(z)=\frac{1}{z(z-{\tfrac{1}{2}})}$ defined on the set
$\mathbb{D}\setminus\\{0,{\tfrac{1}{2}}\\}$. For the pseudomultiplier
$\varphi(z)=\frac{1}{z(z-{\tfrac{1}{2}})}$ on $H^{2}$, we have
$D_{\varphi}=\mathbb{D}\setminus\\{0,{\tfrac{1}{2}}\\}$, $E_{\varphi}$ is the
closed subspace $z(z-{\tfrac{1}{2}})H^{2}$ of $H^{2}$, which has codimension
$2$ in $H^{2}$,and $E_{\varphi}^{\perp}=H^{2}\ominus
z(z-{\tfrac{1}{2}})H^{2}$. The operator
$X_{\varphi}:z(z-{\tfrac{1}{2}})H^{2}\to H^{2}$ is given by
$X_{\varphi}z(z-{\tfrac{1}{2}})f=f$ for all $f\in H^{2}$.

Let us describe the space of ambiguous vectors $\mathcal{A}_{\varphi}$ for
$\varphi$. By Proposition 5.3,

$\mathcal{A}_{\varphi}=E_{\varphi}^{\perp}\cap\ker X_{\varphi}^{*}.$

Consider any $f\in\mathcal{A}_{\varphi}$: then, since $f\in
E_{\varphi}^{\perp}$, $f=c_{1}k_{0}+c_{2}k_{{\tfrac{1}{2}}}$ for some
$c_{1},c_{2}\in\mathbb{C}$, and since $f\in\ker X_{\varphi}^{*}$,
$X_{\varphi}^{*}f=0$. Thus, for every $g\in H^{2}$,

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\left\langle
X_{\varphi}^{*}(c_{1}k_{0}+c_{2}k_{{\tfrac{1}{2}}}),z(z-{\tfrac{1}{2}})g\right\rangle$
(7.5) $\displaystyle=$ $\displaystyle\left\langle
c_{1}k_{0},X_{\varphi}z(z-{\tfrac{1}{2}})g\right\rangle+\left\langle
c_{2}k_{{\tfrac{1}{2}}},X_{\varphi}z(z-{\tfrac{1}{2}})g\right\rangle$
$\displaystyle=$ $\displaystyle c_{1}\left\langle
k_{0},g\right\rangle+c_{2}\left\langle k_{{\tfrac{1}{2}}},g\right\rangle$
$\displaystyle=$ $\displaystyle c_{1}g(0)+c_{2}g({\tfrac{1}{2}}).$

For $g(z)=z$, the equation 7.5 implies that $c_{2}=0$ and, for
$g(z)=z-{\tfrac{1}{2}}$, the equation 7.5 implies that $c_{1}=0$, thus $f=0$.
Therefore $\mathcal{A}_{\varphi}=\\{0\\}$.

Let us describe the set of definable vectors $\mathbf{D}_{\varphi}$ of
$\varphi$. By Lemma 5.1, for $f\in H^{2}$, $\varphi$ sees $f$ with value $c$
if and only if there exists $u\in E^{\perp}_{\varphi}$ such that

$X^{*}_{\varphi}f=\overline{c}f+u,$ (7.6)

equivalently, if and only if there exist $a_{0},a_{1}\in\mathbb{C}$ such that

$X_{\varphi}^{*}f=\bar{c}f+a_{0}k_{0}+a_{1}k_{{\tfrac{1}{2}}}.$

Since $\mathcal{A}_{\varphi}=\\{0\\}$, it is clear that for each
$\overline{c}\not\in\sigma(X^{*}_{\varphi})$,

$f=\left(X_{\varphi}^{*}-\bar{c}I\right)^{-1}(a_{0}k_{0}+a_{1}k_{{\tfrac{1}{2}}}),$

where $a_{0},a_{1}\in\mathbb{C}$ are not all zero, is a definable vector of
$\varphi$. To show that the polar space $\mathcal{P}_{\varphi}$ of $\varphi$
is $\operatorname{span}\\{k_{0},k_{{\tfrac{1}{2}}}\\}$, choose a sequence
$\\{c_{n}\\}\subseteq\mathbb{C}$ with $c_{n}\to\infty$ and
$\overline{c}_{n}\not\in\sigma(X^{*}_{\varphi})$ for all $n$. Then, for any
$g\in E_{\varphi}^{\perp}=\operatorname{span}\\{k_{0},k_{{\tfrac{1}{2}}}\\}$,
the sequence

$v_{n}=\overline{c}_{n}\left(\overline{c}_{n}I-X^{*}_{\varphi}\right)^{-1}g$

is a sequence of definable vectors in $H^{2}$ for $\varphi$, such that
$v_{n}\to g$ and $\varphi(v_{n})\to\infty$ as $n\to\infty$, and so $g$ is a
polar vector of $\varphi$. Therefore,

$\mathcal{P}_{\varphi}=E_{\varphi}^{\perp}=\operatorname{span}\\{k_{0},k_{{\tfrac{1}{2}}}\\}.$

$\square$

######  Remark 7.4.

Question: 1) is the kernel $k_{\lambda}$ visible for all $\lambda\in
D_{\varphi}$?

2) For a given polar vector $u=a_{0}k_{0}+a_{1}k_{{\tfrac{1}{2}}}$ of
$\varphi$, is it the case that $u$ is a limit of definable vectors, each of
which is a sum of two kernels?

######  Example 7.5.

A hybrid 2-dimensional singular space. In this example we conjoin two previous
examples of pseudomultipliers, $\varphi(z)=1/z$ on $H^{2}(\mathbb{D})$ and
$\chi(t)=\sqrt{t}$ on $W^{1,2}[0,1]$ (see examples 2.2 and 2.5) to exhibit a
singular space that contains both polar vectors and ambiguous vectors. Recall
that $D_{\varphi}=\mathbb{D}\setminus\\{0\\}$ and $D_{\chi}=I$, where $I$
denotes the unit interval $I=[0,1]$. Let $\mathbb{D}\sqcup I$ denote the
disjoint union of the topological spaces $\mathbb{D}$ and $I$, and let
$\Omega$ be the topological quotient space of $\mathbb{D}\sqcup I$ obtained by
the identification of the two points $0_{\mathbb{D}}$ and $0_{I}$ in
$\mathbb{D}\sqcup I$ (we distinguish points of $\mathbb{D}\sqcup I$ which
belong both to $\mathbb{D}$ and to $I$ by attaching an appropriate suffix). We
shall denote by $\mathbf{0}$ the point of $\Omega$ obtained by the
identification of the points $0_{\mathbb{D}}$ and $0_{I}$. Let $\mathcal{H}$
be the Hilbert space of functions $\begin{pmatrix}f\\\
g\end{pmatrix}:\Omega\to\mathbb{C}$ defined by

$\begin{pmatrix}f\\\
g\end{pmatrix}(\omega)=\left\\{\begin{array}[]{lcl}f(\omega)&\text{ if
}&\omega=z\in\mathbb{D}\setminus\\{0\\}\\\ g(\omega)&\text{ if }&\omega=t\in
I,\end{array}\right.$

where $f\in H^{2}(\mathbb{D})$ and $g\in W^{1,2}(I)$, equipped with its
natural inner product and operations as a subspace of the Hilbert space direct
sum $H^{2}(\mathbb{D})\oplus W^{1,2}(I)$. It is easy to see that $\mathcal{H}$
is the closed subspace of $\mathcal{K}=H^{2}(\mathbb{D})\oplus W^{1,2}(I)$
comprising the elements $\begin{pmatrix}f\\\ g\end{pmatrix}\in\mathcal{K}$
such that $f(0_{\mathbb{D}})=g(0_{I})$.

Consider the function $\gamma:\Omega\to\mathbb{C}$ defined by

$\gamma(\omega)=\left\\{\begin{array}[]{lcl}\frac{1}{z}&\text{ if
}&\omega=z\in\mathbb{D}\setminus\\{0\\}\\\ \sqrt{t}&\text{ if }&\omega=t\in
I.\end{array}\right.$

We claim that $\gamma$ is a pseudomultiplier of $\mathcal{H}$ on the set
$\Omega$. Furthermore the singular set $\mathcal{S}_{\gamma}$ is
$2$-dimensional and $\mathcal{S}_{\gamma}$ is the orthogonal direct sum of the
one-dimensional spaces $\mathcal{P}_{\gamma}$ and $\mathcal{A}_{\gamma}$.

To prove this statement let us first show that $E_{\gamma}$ has codimension
$2$ in $\mathcal{H}$. For any $\begin{pmatrix}f\\\
g\end{pmatrix}\in\mathcal{H}$, the function $\gamma\begin{pmatrix}f\\\
g\end{pmatrix}$ is given by

$\gamma\begin{pmatrix}f\\\
g\end{pmatrix}(\omega)=\left\\{\begin{array}[]{lcl}\frac{1}{z}f(z)&\text{ if
}&\omega=z\in\mathbb{D}\setminus\\{0\\}\\\ \sqrt{t}g(t)&\text{ if
}&\omega=t\in I.\end{array}\right.$

In particular, $\mathbf{0}\in D_{\gamma}$ and $\gamma(\mathbf{0})=0$. A vector
$\begin{pmatrix}f\\\ g\end{pmatrix}\in\mathcal{H}$ satisfies
$\gamma\begin{pmatrix}f\\\ g\end{pmatrix}\in\mathcal{H}$ if and only if
$\varphi f$ extends to an element $h_{1}$ of $H^{2}$, $\chi g$ extends to an
element $h_{2}$ of $W^{1,2}[0,1]$ and $h_{1}(0_{\mathbb{D}})=h_{2}(0_{I})$.

$E_{\gamma}=\left\\{\begin{pmatrix}f\\\ g\end{pmatrix}\in\mathcal{H}:\text{
there exists }h=\begin{pmatrix}h_{1}\\\ h_{2}\end{pmatrix}\in\mathcal{H}\text{
such that }\gamma\begin{pmatrix}f\\\ g\end{pmatrix}=h|_{D_{\gamma}}\right\\}.$

Hence, by examples 2.2 and 2.5, $f=z\psi$ for some $\psi\in H^{2}$, $g\in
k_{0}^{\perp}$ (here $k_{0}$ is the reproducing kernel of $0\in I$ in
$W^{1,2}(I)$) and

$\begin{pmatrix}h_{1}\\\ h_{2}\end{pmatrix}=\begin{pmatrix}\varphi f\\\ \chi
g\end{pmatrix}=\begin{pmatrix}\psi\\\ \chi g\end{pmatrix}.$ (7.7)

Note that, for $z\in\mathbb{D}$,
$f^{\prime}(z)=(z\psi(z))^{\prime}=\psi(z)+z\psi^{\prime}(z)$, and so, by
equation (7.7), since $\begin{pmatrix}h_{1}\\\
h_{2}\end{pmatrix}\in\mathcal{H}$,

$f^{\prime}(0)=\psi(0)=h_{1}(0)=h_{2}(0)=(\chi g)(0)=0.$

Therefore

$E_{\gamma}=\left\\{\begin{pmatrix}f\\\ g\end{pmatrix}\in\mathcal{H}:f\in
zH^{2},g\in k_{0}^{\perp}\text{ and }f^{\prime}(0)=0\right\\}.$ (7.8)

Let us present $E_{\gamma}$ using reproducing kernels of $\mathcal{H}$. We
shall denote by $s_{z}$ the reproducing kernel of the point $z\in\mathbb{D}$
in the space $H^{2}$, and by $k_{x}$ the reproducing kernel of $x\in I$ in
$W^{1,2}(I)$. We also need the kernel $s^{1}_{z}\in H^{2}$ which reproduces
the derivative at $z$, in the sense that

$\left\langle f,s^{1}_{z}\right\rangle_{H^{2}}=f^{\prime}(z)\text{ for all
}f\in H^{2}.$

Explicitly,

$s^{1}_{z}(\lambda)=\frac{\lambda}{(1-\bar{z}\lambda)^{2}}.$

Let $P_{\mathcal{H}}$ denote the orthogonal projection operator from
$\mathcal{K}$ onto its subspace $\mathcal{H}$. Observe that $\mathcal{H}$ is
the orthogonal complement in $\mathcal{K}$ of the vector
$v=\begin{pmatrix}s_{0}\\\ -k_{0}\end{pmatrix}$, and therefore, for every
$x\in\mathcal{H}$,

$P_{\mathcal{H}}(x)=x-\frac{\left\langle x,v\right\rangle
v}{\|v\|^{2}}=x-\frac{\left\langle x,v\right\rangle v}{1+k_{0}(0)}.$ (7.9)

It is easy to see that the reproducing kernel of the point $z\in\mathbb{D}$ in
$\mathcal{H}$ is $P_{\mathcal{H}}\begin{pmatrix}s_{z}\\\ 0\end{pmatrix}$.
Similarly, the reproducing kernel in $\mathcal{H}$ of the point $x\in I$ is
$P_{\mathcal{H}}\begin{pmatrix}0\\\ k_{x}\end{pmatrix}$. Formula (7.9) for
$P_{\mathcal{H}}$ shows that the reproducing kernel $k_{\mathbf{0}}$ of the
point ${\mathbf{0}}\in\Omega$ in the space $\mathcal{H}$ is given by the
formula

$k_{\mathbf{0}}=P_{\mathcal{H}}\begin{pmatrix}s_{0}\\\
0\end{pmatrix}=P_{\mathcal{H}}\begin{pmatrix}0\\\
k_{0}\end{pmatrix}=\frac{1}{1+k_{0}(0)}\begin{pmatrix}k_{0}(0)s_{0}\\\
k_{0}\end{pmatrix}\;\text{and}\;P_{\mathcal{H}}\begin{pmatrix}s^{1}_{0}\\\
0\end{pmatrix}=\begin{pmatrix}z\\\ 0\end{pmatrix}.$ (7.10)

Re-write equation (7.8) in the form

$E_{\gamma}=\left\\{\begin{pmatrix}f\\\
g\end{pmatrix}\in\mathcal{H}:\begin{pmatrix}f\\\
g\end{pmatrix}\perp\operatorname{span}\left\\{P_{\mathcal{H}}\begin{pmatrix}s_{0}\\\
0\end{pmatrix},P_{\mathcal{H}}\begin{pmatrix}s^{1}_{0}\\\
0\end{pmatrix},P_{\mathcal{H}}\begin{pmatrix}0\\\
k_{0}\end{pmatrix}\right\\}\right\\}.$ (7.11)

Thus, in view of equations (7.10),

$E_{\gamma}=\left\\{\begin{pmatrix}k_{0}(0)s_{0}\\\
k_{0}\end{pmatrix},\begin{pmatrix}z\\\ 0\end{pmatrix}\right\\}^{\perp},$

and so $E_{\gamma}$ has codimension $2$ in $\mathcal{H}$. Therefore $\gamma$
is a pseudomultiplier of $\mathcal{H}$. By equations (7.11) and (7.10)

$\mathcal{S}_{\gamma}=E_{\gamma}^{\perp}=\operatorname{span}\left\\{P_{\mathcal{H}}\begin{pmatrix}s_{0}\\\
0\end{pmatrix},P_{\mathcal{H}}\begin{pmatrix}s^{1}_{0}\\\
0\end{pmatrix},P_{\mathcal{H}}\begin{pmatrix}0\\\
k_{0}\end{pmatrix}\right\\}=\operatorname{span}\left\\{\begin{pmatrix}k_{0}(0)s_{0}\\\
k_{0}\end{pmatrix},\begin{pmatrix}z\\\ 0\end{pmatrix}\right\\}.$ (7.12)

Note that

$\left\langle\begin{pmatrix}k_{0}(0)s_{0}\\\
k_{0}\end{pmatrix},\begin{pmatrix}z\\\
0\end{pmatrix}\right\rangle_{\mathcal{K}}=\left\langle
k_{0}(0)s_{0},z\right\rangle_{H^{2}}+\left\langle
k_{0},0\right\rangle_{W^{1,2}}=0,$

and so equation (7.12) gives an orthogonal basis for $\mathcal{S}_{\gamma}$.
The ambiguous vectors for $\gamma$ are the members of
$E_{\gamma}^{\perp}\cap\ker X_{\gamma}^{*}$. Consider any $\begin{pmatrix}f\\\
g\end{pmatrix}\in\mathcal{A}_{\gamma}$: then $\begin{pmatrix}f\\\
g\end{pmatrix}\in E_{\gamma}^{\perp}$, and so $\begin{pmatrix}f\\\
g\end{pmatrix}=c_{1}\begin{pmatrix}k_{0}(0)s_{0}\\\
k_{0}\end{pmatrix}+c_{2}\begin{pmatrix}z\\\ 0\end{pmatrix}$ for some
$c_{1},c_{2}\in\mathbb{C}$, and $X_{\gamma}^{*}\begin{pmatrix}f\\\
g\end{pmatrix}=0$. Recall that $X_{\gamma}^{*}$ acts from $\mathcal{H}$ to
$E_{\gamma}$. Thus, for every element $h=\begin{pmatrix}h_{1}\\\
h_{2}\end{pmatrix}\in E_{\gamma}$

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\left\langle
X_{\gamma}^{*}\begin{pmatrix}f\\\ g\end{pmatrix},h\right\rangle$
$\displaystyle=$ $\displaystyle\left\langle\begin{pmatrix}f\\\
g\end{pmatrix},X_{\gamma}h\right\rangle.$

In particular, by equation (7.8) we may choose $h_{1}(z)=z^{2}$ and $h_{2}\in
k_{0}^{\perp}$, and so

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\left\langle
c_{1}\begin{pmatrix}k_{0}(0)s_{0}\\\
k_{0}\end{pmatrix}+c_{2}\begin{pmatrix}z\\\
0\end{pmatrix},X_{\gamma}\begin{pmatrix}z^{2}\\\
h_{2}\end{pmatrix}\right\rangle$ $\displaystyle=$ $\displaystyle\left\langle
c_{1}\begin{pmatrix}k_{0}(0)s_{0}\\\
k_{0}\end{pmatrix}+c_{2}\begin{pmatrix}z\\\ 0\end{pmatrix},\begin{pmatrix}z\\\
\chi h_{2}\end{pmatrix}\right\rangle$ $\displaystyle=$ $\displaystyle
c_{1}\left\langle\begin{pmatrix}k_{0}(0)s_{0}\\\
k_{0}\end{pmatrix},\begin{pmatrix}z\\\ \chi
h_{2}\end{pmatrix}\right\rangle+c_{2}\left\langle\begin{pmatrix}z\\\
0\end{pmatrix},\begin{pmatrix}z\\\ \chi h_{2}\end{pmatrix}\right\rangle$
$\displaystyle=$ $\displaystyle c_{1}\left(\left\langle
k_{0}(0)s_{0},z\right\rangle+\left\langle k_{0},\chi
h_{2}\right\rangle\right)+c_{2}\left(\left\langle
z,z\right\rangle+\left\langle 0,\chi h_{2}\right\rangle\right)$
$\displaystyle=$ $\displaystyle
c_{1}k_{0}(0)\overline{z(0)}+c_{1}\overline{\chi(0)h_{2}(0)}+c_{2}\left\langle
z,z\right\rangle$ $\displaystyle=$ $\displaystyle c_{2}\left\langle
z,z\right\rangle=c_{2},$

and so $c_{2}=0$. Hence

$\mathcal{A}_{\gamma}=\mathbb{C}\begin{pmatrix}k_{0}(0)s_{0}\\\
k_{0}\end{pmatrix}=\mathbb{C}P_{\mathcal{H}}\begin{pmatrix}0\\\
k_{0}\end{pmatrix}.$

The singular space of $\gamma$ contains both polar vectors and ambiguous
vectors. $\mathbf{0}\in\Omega$ is an ambiguous point of $\gamma$. By Theorem
7.2, $\mathcal{P}_{\gamma}$ is a closed subspace of $\mathcal{H}$ and

$\mathcal{P}_{\gamma}=\mathcal{S}_{\gamma}\ominus\mathcal{A}_{\gamma}=\operatorname{span}\left\\{\begin{pmatrix}k_{0}(0)s_{0}\\\
k_{0}\end{pmatrix},\begin{pmatrix}z\\\
0\end{pmatrix}\right\\}\ominus\mathbb{C}\begin{pmatrix}k_{0}(0)s_{0}\\\
k_{0}\end{pmatrix}=\mathbb{C}\begin{pmatrix}z\\\ 0\end{pmatrix}.$

$\square$

##  8\. Visible subspaces of a pseudomultiplier

The notion of a pseudomultiplier seeing a vector has proved fruitful. We shall
now take a slightly different approach to that notion which will lead to a
generalization, the idea of a pseudomultiplier seeing a subspace.

An obvious way to generalize Definition 2.7, of $\varphi$ seeing a vector $v$
with value $c$, would be to formalize the situation where knowledge of $n$
moments of $f,\\{\left\langle f,v_{1}\right\rangle,\ \left\langle
f,v_{2}\right\rangle,\dots,\left\langle f,v_{n}\right\rangle\\}$, implies that
the corresponding moments of $X_{\varphi}f,\\{\left\langle