# Model spaces invariant under composition operators

P. Muthukumar  P. Muthukumar, Indian Statistical Institute, Statistics and
Mathematics Unit, 8th Mile, Mysore Road, Bangalore, 560 059, India.
<EMAIL_ADDRESS>and  Jaydeb Sarkar  Jaydeb Sarkar, Indian Statistical
Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore,
560 059, India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Given a holomorphic self-map $\varphi$ of $\mathbb{D}$ (the open unit disc in
$\mathbb{C}$), the composition operator $C_{\varphi}f=f\circ\varphi$, $f\in
H^{2}(\mathbb{\mathbb{D}})$, defines a bounded linear operator on the Hardy
space $H^{2}(\mathbb{\mathbb{D}})$. The model spaces are the backward shift-
invariant closed subspaces of $H^{2}(\mathbb{\mathbb{D}})$, which are
canonically associated with inner functions. In this paper, we study model
spaces that are invariant under composition operators. Emphasis is put on
finite-dimensional model spaces, affine transformations, and linear fractional
transformations.

###### Key words and phrases:

Composition operators, inner functions, model spaces, Hardy space, fractional
linear transformations, Blaschke products.

###### 1991 Mathematics Subject Classification:

Primary: 47B33, 30J05; Secondary: 47B91, 30H10, 30D55, 46E15, 47A15, 46E22,
30J10

##  1\. Preliminaries

Let $\mathbb{D}$ denote the open unit disk in $\mathbb{C}$ and let
$H^{2}(\mathbb{D})$ be the Hardy space over $\mathbb{D}$. For $\varphi$ a
holomorphic self-map of $\mathbb{D}$, define the composition operator
$C_{\varphi}$ on $H^{2}(\mathbb{D})$ by

$C_{\varphi}f=f\circ\varphi\qquad(f\in H^{2}(\mathbb{D})).$

It follows from the Littlewood subordination principle that
$C_{\varphi}\in\mathcal{B}(H^{2}(\mathbb{D}))$ [22]. Here
$\mathcal{B}(H^{2}(\mathbb{D}))$ denote the set of all bounded linear
operators on $H^{2}(\mathbb{D})$.

The theory of composition operators is highly interdisciplinary with its
natural connections to complex analysis, linear dynamics, complex geometry,
and functional analysis. For instance, the question of the existence of
nontrivial invariant subspaces of bounded linear operators on Hilbert spaces
can be formulated in terms of minimal invariant subspaces of composition
operators induced by hyperbolic automorphisms of $\mathbb{D}$ [18, 19, 20].

The theme of this paper is also in line with the invariant subspaces of
composition operators, but restricted to model (or inner function based
quotient) spaces of $H^{2}(\mathbb{D})$. Recall that a function $\theta\in
H^{\infty}(\mathbb{D})$ is said to be inner if $|\theta|=1$ a.e. on
$\partial\mathbb{D}$ in the sense of nontangential boundary values, where
$H^{\infty}(\mathbb{D})$ denotes the commutative Banach algebra of all bounded
holomorphic functions on $\mathbb{D}$. Given an inner function $\theta$, the
model space $\mathcal{Q}_{\theta}$ is the quotient space

$\mathcal{Q}_{\theta}=H^{2}(\mathbb{D})\ominus\theta H^{2}(\mathbb{D})\cong
H^{2}(\mathbb{D})/\theta H^{2}(\mathbb{D}).$

The aim of this paper is to describe some results and methods of the
calculation of model spaces that are invariant under composition operators.
Note, on the other hand, a celebrated result of A. Beurling [1] states that a
closed subspace $\mathcal{Q}\subseteq H^{2}(\mathbb{D})$ is invariant under
$M_{z}^{*}$ if and only if $\mathcal{Q}$ is a model space. Here $M_{z}f=zf$,
$f\in H^{2}(\mathbb{D})$, and $M_{z}^{*}$ is the backward shift

$M_{z}^{*}f=\frac{f(z)-f(0)}{z}\qquad(f\in H^{2}(\mathbb{D})).$

In other words, we are interested in joint invariant subspaces of the (non
commuting) pair $(M_{z}^{*},C_{\varphi})$ on $H^{2}(\mathbb{D})$. Yet another
motivation comes from the recent classification [2] of joint invariant
subspaces of the non commuting pair $(M_{z},C_{\varphi})$ on
$H^{2}(\mathbb{D})$. In this context, we further remark that the concept of
joint invariant subspaces of $(M_{z},C_{\varphi})$ was independently
introduced by Chalendar and Partington [4] and Mahvidi [16]. For recent
results related to the subject we refer to [2, 14, 17] and the references
therein.

Often results on composition operators are case-based. For instance, the
theory of $C_{\varphi}$ corresponding to holomorphic self-map of $\mathbb{D}$
as

$\varphi(z)=az+b\text{ or }\frac{az+b}{cz+d},$

is extremely demanding and complete answers to many basic questions are not
known (cf. [7, 6], and also see [11, 15] for more recent accounts). Even norm
computations and invariant subspaces of $C_{\varphi}$ for linear fractional
transformations are known to be case-based (cf. [3, 18]). In fact, the results
of this paper are also no exception. For example, Theorem 3.1 states: Let
$\varphi$ be a holomorphic self-map of $\mathbb{D}$. Fix $\alpha\in\mathbb{D}$
and $n\in\mathbb{N}$, and consider the inner function

$\theta(z)=\left(\frac{z-\alpha}{1-\bar{\alpha}z}\right)^{n}\qquad(z\in\mathbb{D}).$

Then $\mathcal{Q}_{\theta}$ is invariant under $C_{\varphi}$ if and only if
there exist scalars $a,b$, and $c(\neq 0)$ such that

(1.1) $\varphi(z)=\begin{cases}a+bz&\mbox{if }\alpha=0\text{ and }n>1\\\
\frac{1-c}{\bar{\alpha}}+cz&\mbox{if }\alpha\neq 0,\end{cases}$

where, $|a|+|b|\leq 1$ and $|\frac{1-c}{\bar{\alpha}}|+|c|\leq 1$.

Note that the above model space $\mathcal{Q}_{\theta}$ is an $n$-dimensional
Hilbert space. Even though we all know the structure of finite-dimensional
model spaces (they are essentially parameterized by $n$-Blaschke products) but
we do not know a general method to classify holomorphic self-maps $\varphi$ of
$\mathbb{D}$ for which a given finite-dimensional model space remains
invariant under $C_{\varphi}$ (see Question 3.3). In order to make this
problem more convincing, in Example 3.4 we consider the $2$-dimensional model
space $\mathcal{Q}_{\theta}$ corresponding to the (special) Blaschke product
$\theta(z)=z\frac{z-\alpha}{1-\bar{\alpha}z}$, $z\in\mathbb{D}$, and prove
that $C_{\varphi}\mathcal{Q}_{\theta}\subseteq\mathcal{Q}_{\theta}$ if and
only if $\varphi$ is the special Möbius map

$\varphi(z)=\frac{(c_{1}-1)+(\bar{\alpha}+c_{2})z}{\bar{\alpha}(c_{1}+c_{2}z)}\qquad(z\in\mathbb{D}),$

for some scalars $c_{1}$ and $c_{2}$ such that at least one of them is nonzero
(see (3.2)). The above answers do not seem to tell us much about a possible
answer to our general question.

We also put special emphasis on composition operators induces by linear
fractional symbols (cf. Theorem 4.5): Let $\theta$ be an inner function, and
let $\varphi(z)=\frac{az+b}{cz+d}$ be a nonconstant holomorphic self-map of
$\mathbb{D}$, where $ad-bc=1$. Suppose
$\sigma(z)=\frac{\bar{a}z-\bar{c}}{-\bar{b}z+\bar{d}}$ and $\theta(0)=0$. If
we denote $\psi=(\frac{\theta}{z}\circ\sigma)(\frac{\theta}{z})^{-1}$, then
the following are equivalent:

  1. (1)

$\mathcal{Q}_{\theta}$ is invariant under $C_{\varphi}$.

  2. (2)

$\psi\in H^{2}(\mathbb{D})$ or $\psi\in H^{\infty}(\mathbb{D})$.

  3. (3)

$\frac{\theta}{z}H^{2}(\mathbb{D})$ is invariant under $C_{\sigma}$.

A similar (but not same as above) consideration also holds for composition
operators induced by affine transformations (see Theorem 4.3).

In the application part, we consider direct interpretations of our results to
model spaces and reducing subspaces of composition operators. For instance,
Theorem 5.3 states that:

  1. (1)

$\mathcal{Q}_{\theta}$ is invariant under $C_{\varphi}$ for all inner function
$\theta$ if and only if $\varphi$ is the identity map.

  2. (2)

$\mathcal{Q}_{\theta}$ is invariant under $C_{\varphi}$ for all holomorphic
self-map $\varphi$ if and only if $\theta(z)=\alpha z$ or
$\theta\equiv\alpha$, where $\alpha$ is an unimodular constant.

We now discuss the issue of reducing subspaces of composition operators from
our perspective. Let $T$ be a bounded linear operator on a Hilbert space
$\mathcal{H}$. A closed subspace $\mathcal{M}\subseteq\mathcal{H}$ is called
reducing subspace for $T$ (or $\mathcal{M}$ reduces $T$) if both $\mathcal{M}$
and $\mathcal{M}^{\bot}$ are invariant under $T$. It is known that $M_{z}$ on
$H^{2}(\mathbb{D})$ is irreducible, that is, $M_{z}$ has no nontrivial
reducing subspaces. In the setting of composition operators, if
$\alpha\in\mathbb{C}$ with $\text{Re}(\alpha)>0$, then $C_{\varphi_{\alpha}}$
is also irreducible [18, Theorem 1.3], where

$\varphi_{\alpha}(z)=\frac{(2-\alpha)z+\alpha}{-\alpha
z+(2+\alpha)}\qquad(z\in\mathbb{D}).$

Another example is the composition operator corresponding to univalent
loxodromic type self-map with nonzero Denjoy-Wolff point [13, Theorem 1]. At
the other extreme, we discuss reducibility of compositor operators restricted
to model spaces. In Theorem 5.5, we prove the following: Let $\varphi$ be a
holomorphic self-map of $\mathbb{D}$, $\alpha\in\mathbb{D}$, $n\geq 1$, and
suppose $\theta(z)=\left(\frac{z-\alpha}{1-\bar{\alpha}z}\right)^{n}$. Then
$\mathcal{Q}_{\theta}$ reduces $C_{\varphi}$ if and only if there exist a
scalar $c$, $|c|\leq 1$, and $\psi\in H^{\infty}(\mathbb{D})$,
$\|\psi\|_{\infty}\leq 1$, such that

$\varphi(z)=\begin{cases}z\psi&\mbox{if }\alpha=0\text{ and }n=1\\\
cz&\mbox{if }\alpha=0\text{ and }n\geq 2\\\ z&\mbox{if }\alpha\neq
0.\end{cases}$

Finally, we make a general remark. Recall that a linear fractional
transformation $\varphi(z)=\frac{az+b}{cz+d}$ is a self-map of $\mathbb{D}$ if
and only if

$|b\bar{d}-a\bar{c}|+|ad-bc|\leq|d|^{2}-|c|^{2}.$

In particular, since $\varphi$ is a holomorphic self-map, the inequalities
following (1.1) are automatically true. In this paper, whenever no confusion
can arise, we will not explicitly mention the above inequality for linear
fractional transformations that are self-maps of $\mathbb{D}$. For instance,
the above inequality is applicable to Theorem 3.1, Example 3.4, and Theorem
4.3.

The rest of the paper is organized as follows. Section 2 recalls (rather a
modified version of) the classification of Beurling type invariant subspaces
from [2]. Section 3 relates affine transformations and certain finite-
dimensional model spaces. Section 4 deals with model subspaces invariant under
composition operators corresponding to fractional linear transformations. The
final section, Section 5, deals with some of the direct applications of our
main results. We also discuss irreducible composition operators at the level
of model spaces.

##  2\. Beurling type invariant subspaces

We begin with the formal definition of Hardy space $H^{2}(\mathbb{D})$. Recall
that the Hardy space is the function Hilbert space of all holomorphic
functions on $\mathbb{D}$ whose power series have square-summable
coefficients. The following equivalent description of $H^{2}(\mathbb{D})$ is
well known:

$H^{2}(\mathbb{D})=\Big{\\{}f\in\text{Hol}(\mathbb{D}):\|f\|:=\Big{[}\sup_{0\leq
r<1}\frac{1}{2\pi}\int_{0}^{2\pi}|f(re^{it})|^{2}dt\Big{]}^{\frac{1}{2}}<\infty\Big{\\}}.$

Also we denote the closed unit ball of $H^{\infty}(\mathbb{D})$ (which is
popularly known as Schur class) by

${\mathcal{S}}(\mathbb{D})=\\{\psi\in
H^{\infty}(\mathbb{D}):\|\psi\|_{\infty}:=\sup_{z\in\mathbb{D}}|\psi(z)|\leq
1\\}.$

We follow the standard notation: Given a bounded linear operator $T$ on some
Hilbert space $\mathcal{H}$, the lattice of all closed $T$-invariant subspaces
is denoted by $\text{Lat}T$, that is

$\text{Lat}T=\\{\mathcal{S}\subseteq\mathcal{H}\text{ closed
subspace}:T\mathcal{S}\subseteq\mathcal{S}\\}.$

By Beurling’s Theorem, we know that

$\text{Lat}M_{z}=\\{\\{0\\},\theta H^{2}(\mathbb{D}):\theta\text{
inner}\\},\text{ and
}\text{Lat}M_{z}^{*}=\\{H^{2}(\mathbb{D}),\mathcal{Q}_{\theta}:\theta\text{
inner}\\}.$

A closed subspace $\mathcal{S}\subseteq H^{2}(\mathbb{D})$ is said to be a
Beurling type invariant subspace if $\mathcal{S}=\theta H^{2}(\mathbb{D})$ for
some inner function $\theta$, or equivalently (in view of Beurling),
$\mathcal{S}$ is invariant under $M_{z}$. The purpose of this short section is
to recall the classification of Beurling type invariant subspaces of
composition operators [2, Theorem 2.3]. However, the present version is a
simple refinement of [2], and we only prove the new component of the result.

######  Theorem 2.1.

Let $\theta$ be an inner function and $\varphi$ be a holomorphic self-map of
$\mathbb{D}$. The following are equivalent:

  1. (1)

$\theta H^{2}(\mathbb{D})\in\text{Lat}C_{\varphi}$.

  2. (2)

$\displaystyle\frac{\theta\circ\varphi}{\theta}\in{\mathcal{S}}(\mathbb{D})$.

  3. (3)

$\displaystyle\frac{\theta\circ\varphi}{\theta}\in H^{\infty}(\mathbb{D})$.

  4. (4)

$\displaystyle\frac{\theta\circ\varphi}{\theta}\in H^{2}(\mathbb{D})$.

Proof. The equivalence $(1)\Leftrightarrow(2)$ follows from [2, Theorem 2.3].
Since

${\mathcal{S}}(\mathbb{D})\subsetneq H^{\infty}(\mathbb{D})\subsetneq
H^{2}(\mathbb{D}),$

it follows that $(2)\Rightarrow(3)\Rightarrow(4)$. Therefore, it suffices to
verify that $(4)\Rightarrow(2)$. Suppose that
$\frac{\theta\circ\varphi}{\theta}\in H^{2}(\mathbb{D})$. Since $\theta$ is
inner, we have

$\left|\frac{\theta\circ\varphi}{\theta}(e^{it})\right|=|(\theta\circ\varphi)(e^{it})|\leq
1\mbox{~{}for all~{}}t\in[0,2\pi]\mbox{~{}a.e.}$

Therefore, $\frac{\theta\circ\sigma}{\theta}\in H^{\infty}(\mathbb{D})$ with
$\left\|\frac{\theta\circ\varphi}{\theta}\right\|_{\infty}\leq 1$ [9, Theorem
2.11] , that is,
$\frac{\theta\circ\varphi}{\theta}\in{\mathcal{S}}(\mathbb{D})$, which
completes the proof of the theorem. ∎

This result will play a key role in what follows.

##  3\. Finite-dimensional model spaces

Recall that the Hardy space $H^{2}(\mathbb{D})$ is associated with the Szegö
kernel $k:\mathbb{D}\times\mathbb{D}\rightarrow\mathbb{C}$, where

$k(z,w)=(1-z\bar{w})^{-1}\qquad(z,w\in\mathbb{D}).$

Therefore, the linear span of $\\{k(\cdot,w):w\in\mathbb{D}\\}$ is dense in
$H^{2}(\mathbb{D})$, and $f(w)=\langle
f,k(\cdot,w)\rangle_{H^{2}(\mathbb{D})}$ for all $f\in H^{2}(\mathbb{D})$ and
$w\in\mathbb{D}$. For each $\alpha\in\mathbb{D}$, define the Blaschke factor
$b_{\alpha}$ by

$b_{\alpha}(z)=\frac{z-\alpha}{1-\bar{\alpha}z}\qquad(z\in\mathbb{D}).$

It is known that a model space $\mathcal{Q}_{\theta}$ is finite dimensional if
and only if $\theta$ is a finite Blaschke product (cf. [21, Theorem 3.8]).
More specifically, if $\text{dim}\mathcal{Q}_{\theta}=n<\infty$, then there
exists $\alpha_{1},\ldots,\alpha_{m}\in\mathbb{D}$, and
$n_{1},\ldots,n_{m}\in\mathbb{N}$ such that $\sum_{i=1}^{m}n_{i}=n$ and

$\theta=\prod_{i=1}^{m}b_{\alpha_{i}}^{n_{i}}.$

Moreover, $\\{c_{\alpha_{i}}^{(l_{i})}:0\leq l_{i}\leq n_{i}-1,1\leq i\leq
m\\}$ is a basis for $\mathcal{Q}_{\theta}$ (see [12, Proposition 5.16]),
where

$c_{\alpha}^{(t)}:=\frac{z^{t}}{(1-\bar{\alpha}z)^{t+1}}\qquad(t\in\mathbb{Z}_{+}).$

In the following, we prove that a “simple” finite dimensional models space is
invariant under $C_{\varphi}$ if and only if $\varphi$ is a special affine
map.

######  Theorem 3.1.

Let $\varphi$ be a holomorphic self-map of $\mathbb{D}$,
$\alpha\in\mathbb{D}$, $n\in\mathbb{N}$, and suppose
$\theta(z)=\left(\frac{z-\alpha}{1-\bar{\alpha}z}\right)^{n}$. Then
$\mathcal{Q}_{\theta}\in\text{Lat}C_{\varphi}$ if and only if there exist
scalars $a,b$, and $c(\neq 0)$ such that

$\varphi(z)=\begin{cases}a+bz&\mbox{if }\alpha=0\text{ and }n>1\\\
\frac{1-c}{\bar{\alpha}}+cz&\mbox{if }\alpha\neq 0.\end{cases}$

Proof. In view of the preceding discussion, we have

$\mathcal{Q}_{\theta}=\text{span}\left\\{\frac{p(z)}{(1-\bar{\alpha}z)^{n}}:p\in\mathbb{C}[z],\text{deg
}p\leq n-1\right\\}.$

Suppose $\alpha=0$. Then $\theta(z)=z^{n}$ and

$\mathcal{Q}_{\theta}=\mbox{span}\\{1,z,z^{2},\ldots,z^{n-1}\\}.$

Let $\varphi=a+bz$. Clearly, if
$f=\sum_{j=0}^{n-1}a_{j}z^{j}\in\mathcal{Q}_{\theta}$, then

$f\circ\varphi=\sum_{j=0}^{n-1}a_{j}\varphi^{j}=\sum_{j=0}^{n-1}a_{j}(a+bz)^{j},$

and hence $C_{\varphi}f=f\circ\varphi\in\mathcal{Q}_{\theta}$. For the
converse part, suppose
$C_{\varphi}\mathcal{Q}_{\theta}\subseteq\mathcal{Q}_{\theta}$. Since the
identity map $f(z)=z$, $z\in\mathbb{D}$, is in $\mathcal{Q}_{\theta}$, if
follows that

$f\circ\varphi=\varphi\in\mathcal{Q}_{\theta},$

which implies that $\varphi$ is a polynomial. Also note that
$g\in\mathcal{Q}_{\theta}$, where $g(z)=z^{n-1}$, $z\in\mathbb{D}$. Then

$g\circ\varphi=\varphi^{n-1}\in\mathcal{Q}_{\theta},$

which implies that

$(n-1)\mbox{deg}(\varphi)=\mbox{deg}(\varphi^{n-1})\leq n-1.$

Since $n-1\neq 0$, it follows that $\mbox{deg}(\varphi)\leq 1$, and hence
$\varphi$ is an affine map. Now suppose that $\alpha\neq 0$, and let
$C_{\varphi}\mathcal{Q}_{\theta}\subseteq\mathcal{Q}_{\theta}$. Since
$k(\cdot,\alpha)\in\mathcal{Q}_{\theta}$, it follows that

$k(\cdot,\alpha)\circ\varphi=\displaystyle\frac{1}{1-\bar{\alpha}\varphi}\in\mathcal{Q}_{\theta},$

and hence, there exists a polynomial $p$ of degree $\leq n-1$ such that

$\frac{1}{1-\bar{\alpha}\varphi}=\frac{p(z)}{(1-\bar{\alpha}z)^{n}}.$

In particular, $\varphi$ is a nonconstant polynomial. Suppose
$\mbox{deg}(\varphi)=m>0$. Since

$g=\frac{1}{(1-\bar{\alpha}z)^{n}}\in\mathcal{Q}_{\theta},$

we have

$g\circ\varphi=\frac{1}{(1-\bar{\alpha}\varphi)^{n}}\in\mathcal{Q}_{\theta}.$

Thus, there exists $q\in\mathbb{C}[z]$ such that

$(1-\bar{\alpha}\varphi)^{n}q(z)=(1-\bar{\alpha}z)^{n}.$

Comparing the degrees of both sides, we have $mn\leq n$, which implies
$\mbox{deg}(\varphi)=1$. Then $\mbox{deg}(q)=0$, and hence $q$ is a nonzero
constant. Therefore, there exists a nonzero scalar $c$ such that
$1-\bar{\alpha}\varphi=c(1-\bar{\alpha}z)$, that is

$\varphi(z)=\frac{1-c}{\bar{\alpha}}+cz.$

We now turn to the converse direction. Let $c\neq 0$ and
$\varphi(z)=\frac{1-c}{\bar{\alpha}}+cz$. Then
$1-\bar{\alpha}\varphi=c(1-\bar{\alpha}z)$, and hence

$\left(\frac{1}{1-\bar{\alpha}\varphi}\right)^{l}\in\mbox{span}\left\\{\frac{1}{(1-\bar{\alpha}z)^{j}}:j=1,\ldots,n\right\\}=\mathcal{Q}_{\theta}$

for all $l=1,2,\ldots,n$. It follows that
$f\circ\varphi\in\mathcal{Q}_{\theta}$ for all $f\in\mathcal{Q}_{\theta}$.
This completes the proof of the theorem. ∎

We would like to point out that even affine transformations (as symbols) play
a prominent role in the complexity of composition operators (cf. Deddens [7]).

######  Remark 3.2.

In the context of Theorem 3.1, we note that if $\alpha=0$ and $n=1$, then
$\mathcal{Q}_{z}=(zH^{2}(\mathbb{D}))^{\bot}$, which is the one dimensional
subspace of all constant functions in $H^{2}(\mathbb{D})$. Clearly,
$\mathcal{Q}_{z}$ is invariant under every composition operator.

We do not know the answer to the following compelling question.

######  Question 3.3.

Consider the finite Blaschke product $\prod_{i=1}^{m}b_{\alpha_{i}}$
corresponding to $\alpha_{1},\ldots,\alpha_{n}\in\mathbb{D}$. Characterize
holomorphic self-maps $\varphi$ of $\mathbb{D}$ such that

$C_{\varphi}\mathcal{Q}_{\prod_{i=1}^{m}b_{\alpha_{i}}}\subseteq\mathcal{Q}_{\prod_{i=1}^{m}b_{\alpha_{i}}}.$

We also put forward the following simpler (two-dimensional) version: Suppose
$\alpha\neq\beta$ in $\mathbb{D}$, and suppose $\theta=b_{\alpha}b_{\beta}$.
Classify holomorphic self-maps $\varphi$ of $\mathbb{D}$ such that
$C_{\varphi}\mathcal{Q}_{\theta}\subseteq\mathcal{Q}_{\theta}$.

We do not know the complete answer to this question but the following special
case ($\alpha\neq 0$ and $\beta=0$):

######  Example 3.4.

Let $\alpha\in\mathbb{D}$ be a nonzero scalar, and suppose
$\theta=zb_{\alpha}$. Let $\varphi$ be a holomorphic self-map of $\mathbb{D}$.
Then $C_{\varphi}\mathcal{Q}_{\theta}\subseteq\mathcal{Q}_{\theta}$ if and
only if $\varphi$ is the special Möbius map

(3.2)
$\varphi(z)=\frac{(c_{1}-1)+(\bar{\alpha}+c_{2})z}{\bar{\alpha}(c_{1}+c_{2}z)}\qquad(z\in\mathbb{D}),$

for some scalars $c_{1}$ and $c_{2}$ such that at least one of them is
nonzero. Indeed, here the $2$-dimensional model space $\mathcal{Q}_{\theta}$
is given by $\mathcal{Q}_{\theta}=\text{span}\\{1,k(\cdot,\alpha)\\}$. As the
space of all constant functions is invariant under every composition operators
(see Remark 3.2), $\mathcal{Q}_{\theta}$ is invariant under $C_{\varphi}$ if
and only if

$k(\cdot,\alpha)\circ\varphi=\frac{1}{1-\bar{\alpha}\varphi}\in\mathcal{Q}_{\theta}.$

This is equivalent to the fact that

$\frac{1}{1-\bar{\alpha}\varphi}=\frac{c_{1}+c_{2}z}{1-\bar{\alpha}z}\qquad(z\in\mathbb{D}),$

for some scalars $c_{1}$ and $c_{2}$ such that at least one of them is
nonzero, which is equivalent to (3.2). This completes the proof of the claim.

Putting together the representations of linear fractional transformations from
(3.2) and Theorem 3.1, it is perhaps evident that the answer to Question 3.3
will be also case-based.

##  4\. Linear fractional symbols

Our aim in this section is to describe the structure of model invariant
subspaces of composition operators whose symbols are linear fractional
transformations mapping $\mathbb{D}$ into itself. We are also motivated by the
intricate analysis presented in Cowen [6] and Deddens [7].

Before we begin with a general observation, we introduce the following
(standard) notation: For each $\varphi\in H^{\infty}(\mathbb{D})$, we denote
by $M_{\varphi}$ the multiplication operator $M_{\varphi}f=\varphi f$, $f\in
H^{2}(\mathbb{D})$. It is well known that
$M_{\varphi}\in\mathcal{B}(H^{2}(\mathbb{D}))$ with
$\|M_{\varphi}\|=\|\varphi\|_{\infty}$ for all $\varphi\in
H^{\infty}(\mathbb{D})$, and $\\{M_{z}\\}^{\prime}=\\{M_{\varphi}:\varphi\in
H^{\infty}(\mathbb{D})\\}$, where $\\{M_{z}\\}^{\prime}$ denotes the
commutator of $M_{z}$.

Let $\varphi$ be a holomorphic self-map of $\mathbb{D}$ and let $\theta$ be an
inner function. Clearly,
$C_{\varphi}\mathcal{Q}_{\theta}\subseteq\mathcal{Q}_{\theta}$ if and only if
$C_{\varphi}^{*}(\theta H^{2}(\mathbb{D}))\subseteq\theta H^{2}(\mathbb{D})$,
which is, by the Douglas range inclusion theorem [8, Theorem 1] (as
$\mbox{ran}(C_{\varphi}^{*}M_{\theta})\subseteq\mbox{ran}M_{\theta}$),
equivalent to the existence of some $X\in\mathcal{B}(H^{2}(\mathbb{D}))$ such
that

$C_{\varphi}^{*}M_{\theta}=M_{\theta}X.$

Now if the above equality holds, then $\theta X(f)=C_{\varphi}^{*}(\theta f)$
for all $f\in H^{2}(\mathbb{D})$, and hence zeros of $\theta$ are also zeros
of $C_{\varphi}^{*}(\theta f)$ and

$\text{multiplicity}_{w}\theta\leq\text{multiplicity}_{w}C_{\varphi}^{*}(\theta
f),$

for all $w\in\mathcal{Z}(\theta)$ (the zero set of $\theta$). This implies
that $\frac{1}{\theta}C_{\varphi}^{*}(\theta f)$ is a well defined holomorphic
function on $\mathbb{D}$. Therefore, we have proved the following general (but
perhaps less practical) characterization.

######  Lemma 4.1.

$Q_{\theta}$ is invariant under $C_{\varphi}$ if and only if $f\in
H^{2}(\mathbb{D})\mapsto\frac{1}{\theta}C_{\varphi}^{*}M_{\theta}f$ defines a
bounded linear operator on $H^{2}(\mathbb{D})$.

Now we turn to linear fractional transformations. The following adjoint
formula is due to Cowen [6, Theorem 2]. There is a wide range of applications
of Cowen’s adjoint formula (cf. [5, 10, 11]).

######  Theorem 4.2.

Let $\varphi(z)=\frac{az+b}{cz+d}$ be a nonconstant holomorphic self-map of
$\mathbb{D}$, where $ad-bc=1$. Then
$\sigma(z)=\frac{\bar{a}z-\bar{c}}{-\bar{b}z+\bar{d}}$ is a holomorphic self-
map of $\mathbb{D}$, $g(z)=\frac{1}{-\bar{b}z+\bar{d}}$ and $h(z)=cz+d$ are in
$H^{\infty}(\mathbb{D})$, and $C_{\varphi}^{*}=M_{g}C_{\sigma}M_{h}^{*}$.

Note that if $\varphi$ is a constant map, then $\mathcal{Q}_{\theta}$ is
invariant for $C_{\varphi}$ if and only if $\theta(0)=0$. Indeed, the constant
function $1\in\mathcal{Q}_{\theta}$ if and only if $\theta(0)=\langle
1,\theta\rangle_{H^{2}(\mathbb{D})}=0$. Now we turn to nonconstant affine
self-maps.

In what follows, we denote by $\mathcal{H}(\mathbb{D})$ either
$H^{2}(\mathbb{D})$ or $H^{\infty}(\mathbb{D})$.

######  Theorem 4.3.

Let $\theta$ be an inner function, and let $\varphi(z)=az+b,a\neq 0$. Suppose
$\sigma(z)=\frac{\bar{a}z}{1-\bar{b}z}$, $z\in\mathbb{D}$. The following are
equivalent.

  1. (1)

$\mathcal{Q}_{\theta}\in\text{Lat}C_{\varphi}$.

  2. (2)

$\frac{\theta\circ\sigma}{\theta}\in\mathcal{H}(\mathbb{D})$.

  3. (3)

$\theta H^{2}(\mathbb{D})\in\text{Lat}C_{\sigma}$.

Proof. If we apply Theorem 4.2 to the case when $c=0$ and $d=1$, we get

$C_{\varphi}^{*}=M_{\psi}C_{\sigma},$

where $\psi(z)=\frac{1}{1-\bar{b}z}$. Now suppose that $\mathcal{Q}_{\theta}$
is invariant under $C_{\varphi}$. By Lemma 4.1, we know that the map

$Xf=\frac{1}{\theta}C_{\varphi}^{*}M_{\theta}f\qquad(f\in H^{2}(\mathbb{D})),$

defines a bounded linear operator $X$ on $H^{2}(\mathbb{D})$, and hence, using
$C_{\varphi}^{*}=M_{\psi}C_{\sigma}$, we have

$Xf=\frac{1}{\theta}C_{\varphi}^{*}M_{\theta}f=\frac{1}{\theta}\psi(\theta\circ\sigma)(f\circ\sigma)\in
H^{2}(\mathbb{D}),$

for all $f\in H^{2}(\mathbb{D})$. Applying this with $f=1$ yields

$\psi\frac{\theta\circ\sigma}{\theta}\in H^{2}(\mathbb{D}).$

But since $\frac{1}{\psi}=1-\bar{b}z\in H^{\infty}(\mathbb{D})$, it follows
that $\frac{\theta\circ\sigma}{\theta}\in H^{2}(\mathbb{D})$. Theorem 2.1 then
implies that $\theta H^{2}(\mathbb{D})$ is invariant under $C_{\sigma}$. This
proves $(1)\Rightarrow(2)\Rightarrow(3)$.

Now we assume that $C_{\sigma}(\theta H^{2}(\mathbb{D}))\subseteq\theta
H^{2}(\mathbb{D})$. Again, Theorem 2.1 implies that
$\frac{\theta\circ\sigma}{\theta}\in{\mathcal{S}}(\mathbb{D})$, and therefore
$\psi\frac{\theta\circ\sigma}{\theta}\in H^{\infty}(\mathbb{D})$. Then
$\frac{1}{\theta}C_{\varphi}^{*}M_{\theta}$ defines a bounded linear operator
on $H^{2}(\mathbb{D})$, and hence, $\mathcal{Q}_{\theta}$ is invariant under
$C_{\varphi}$. This proves that $(3)\Rightarrow(2)\Rightarrow(1)$ and
completes the proof of the theorem. ∎

Now we consider model spaces that are invariant under composition operators
induced by linear fractional transformations. However, we will be restricted
to inner functions that vanish at the origin. In fact, our main tool is the
following adjoint formula [23, Equation (3.6)], which concerns functions
vanishing at the origin:

######  Lemma 4.4.

Let $\varphi(z)=\frac{az+b}{cz+d}$ be a nonconstant holomorphic self-map of
$\mathbb{D}$, where $ad-bc=1$. If
$\sigma(z)=\frac{\bar{a}z-\bar{c}}{-\bar{b}z+\bar{d}}$, then

$C_{\varphi}^{*}f(z)=z\sigma^{\prime}(z)\frac{f(\sigma(z))}{\sigma(z)}\qquad(f\in
H^{2}(\mathbb{D}),f(0)=0).$

As pointed our earlier, if $\theta$ is an inner function, then the constant
function $1\in\mathcal{Q}_{\theta}$ if and only if $\theta(0)=0$. In the
following, under the assumption that the inner functions vanish at the origin,
we present a classification of model spaces that are invariant under
composition operators induced by linear fractional transformations. Recall
that $\mathcal{H}(\mathbb{D})$ is either $H^{2}(\mathbb{D})$ or
$H^{\infty}(\mathbb{D})$.

######  Theorem 4.5.

Let $\theta$ be an inner function, $\theta(0)=0$, and let
$\varphi(z)=\frac{az+b}{cz+d}$ be a nonconstant holomorphic self-map of
$\mathbb{D}$, where $ad-bc=1$. Suppose
$\sigma(z)=\frac{\bar{a}z-\bar{c}}{-\bar{b}z+\bar{d}}$ and

$\psi=\frac{\frac{\theta}{z}\circ\sigma}{\frac{\theta}{z}}.$

The following are equivalent:

  1. (1)

$\mathcal{Q}_{\theta}\in\text{Lat}C_{\varphi}$.

  2. (2)

$\psi\in\mathcal{H}(\mathbb{D})$.

  3. (3)

$\frac{\theta}{z}H^{2}(\mathbb{D})\in\text{Lat}C_{\sigma}$.

Proof. Since $\theta(0)=0$, we see that $\frac{\theta}{z}$ is an inner
function. Suppose that $\mathcal{Q}_{\theta}$ is invariant under
$C_{\varphi}$. Lemma 4.1 implies that

$Xf=\frac{1}{\theta}C_{\varphi}^{*}(\theta f)\qquad(f\in H^{2}(\mathbb{D})),$

defines a bounded linear operator $X$ on $H^{2}(\mathbb{D})$ and
$C_{\varphi}^{*}M_{\theta}=M_{\theta}X$. In view of Lemma 4.4, it then follows
that

$Xf=\sigma^{\prime}\frac{z(\theta\circ\sigma)}{\theta\sigma}(f\circ\sigma)=\sigma^{\prime}\psi(f\circ\sigma),$

where $\sigma^{\prime}$ is the derivative of $\sigma$. In particular

$X1=\sigma^{\prime}\psi\in H^{2}(\mathbb{D}),$

from which, using the fact that $\frac{1}{\sigma^{\prime}}\in
H^{\infty}(\mathbb{D})$, it follows that $\psi\in H^{2}(\mathbb{D})$. And
finally, by Theorem 2.1, $\displaystyle\frac{\theta}{z}H^{2}(\mathbb{D})$ is
invariant under $C_{\sigma}$, which proves $(1)\Rightarrow(2)\Rightarrow(3)$.

Now if $(3)$ is true, then Theorem 2.1 implies that $(2)$ is true, that is,
$\psi\in H^{\infty}(\mathbb{D})$. In particular, since $\sigma^{\prime}\in
H^{\infty}(\mathbb{D})$, $X$ (defined as above) defines a bounded linear
operator on $H^{2}(\mathbb{D})$ and satisfy the equality
$C_{\varphi}^{*}M_{\theta}=M_{\theta}X$. By Lemma 4.1, $\mathcal{Q}_{\theta}$
is invariant under $C_{\varphi}$, which complete the proof of
$(2)\Rightarrow(1)$. ∎

We do not know, in general, how to classify an inner function $\theta$ for
which $\mathcal{Q}_{\theta}$ is invariant under a composition operator induced
by a linear fractional transformation.

##  5\. Applications

The main goal of this section is to discuss some direct applications of our
main results to some elementary invariant subspaces of composition operators.

First, we prove that the identity map is the only self-map of $\mathbb{D}$,
which leaves all model spaces invariant.

######  Theorem 5.1.

Let $\varphi$ be a holomorphic self-map of $\mathbb{D}$. Then $\theta
H^{2}(\mathbb{D})$ is invariant under $C_{\varphi}$ for all inner function
$\theta$ if and only if $\varphi$ is the identity map.

Proof. If $\varphi(z)=z$, $z\in\mathbb{D}$, then
$C_{\varphi}=I_{H^{2}(\mathbb{D})}$ (the identity operator). In particular,
$\theta H^{2}(\mathbb{D})$ is invariant under $C_{\varphi}$. Conversely,
suppose $\theta H^{2}(\mathbb{D})\in\mbox{Lat}C_{\varphi}$ for all inner
function $\theta$. Fix $\alpha\in\mathbb{D}$. Then $\alpha$ is the only zero
of the Blaschke factor (which is also an inner function)
$b_{\alpha}(z)=\frac{\alpha-z}{1-\bar{\alpha}z}$, $z\in\mathbb{D}$. Since
$b_{\alpha}H^{2}(\mathbb{D})\in\mbox{Lat}C_{\varphi}$, by assumption, Theorem
2.1 implies that

$\frac{b_{\alpha}\circ\varphi}{b_{\alpha}}\in{\mathcal{S}}(\mathbb{D}).$

This forces $b_{\alpha}(\varphi(\alpha))=0$, and hence
$\varphi(\alpha)=\alpha$. Since $\alpha\in\mathbb{D}$ is arbitrary, we have
that $\varphi(z)=z$, $z\in\mathbb{D}$, which completes the proof. ∎

Before examining the other extreme property of the invariant subspaces of
composition operators, we need to recall the following fact which concerns
fixed points of holomorphic self-maps [2, Corollary 3.4]: Let $\varphi$ be a
holomorphic self-map of $\mathbb{D}$ and let $w\in\mathbb{D}$ be the fixed
point of $\varphi$. If $\theta$ is an inner function and $\theta(w)\neq 0$,
then $\theta H^{2}(\mathbb{D})$ is invariant under $C_{\varphi}$ if and only
if $\theta\circ\varphi=\theta$.

Recall that a point $w\in\mathbb{D}$ is said to be a fixed point of a
holomorphic self-map $\varphi$ of $\mathbb{D}$ if $\varphi(w)=w.$

######  Theorem 5.2.

Let $\theta$ be an inner function. Then $\theta
H^{2}(\mathbb{D})\in\text{Lat}C_{\varphi}$ for all self-maps $\varphi$ of
$\mathbb{D}$ if and only if $\theta$ is an unimodular constant.

Proof. If $\theta$ is an unimodular constant, then
$\frac{\theta\circ\varphi}{\theta}$ is also unimodular constant. Theorem 2.1
then implies that $\theta H^{2}(\mathbb{D})\in\mbox{Lat}C_{\varphi}$.

Conversely, assume that $\theta H^{2}(\mathbb{D})\in\text{Lat}C_{\varphi}$ for
all holomorphic self-map $\varphi$ of $\mathbb{D}$. First, we claim that
$\theta$ has no zeros in $\mathbb{D}$. Suppose towards a contradiction that
$\theta(\alpha)=0$ for some $\alpha\in\mathbb{D}$. Since $\theta$ is an inner
function, there exists $\beta\in\mathbb{D}$ such that $\theta(\beta)\neq 0$.
Set

$\varphi=b_{\beta}\circ b_{\alpha}.$

Note that $\varphi$ is an inner function and $\varphi(\alpha)=\beta$.
Therefore, $\frac{\theta\circ\varphi}{\theta}\notin H^{\infty}(\mathbb{D})$,
and hence, by Theorem 2.1, $\theta H^{2}(\mathbb{D})$ is not invariant under
$C_{\varphi}$. This contradiction completes the proof of the claim that
$\theta(z)\neq 0$ for all $z\in\mathbb{D}$. If we consider a constant function
$\varphi\equiv c$, then in view of the discussion preceding the statement of
this theorem, $C_{\varphi}(\theta H^{2}(\mathbb{D}))\subseteq\theta
H^{2}(\mathbb{D})$ implies that

$\theta=\theta\circ\varphi\equiv\theta(c).$

Since $\theta$ is an inner function, $\theta(c)$ must be unimodular, and hence
$\theta$ is an unimodular constant. This completes the proof of the theorem. ∎

Now we turn to special model invariant subspaces of composition operators. In
the following, we will let $\theta$ and $\varphi$ denote inner function and
holomorphic self-map of $\mathbb{D}$, respectively.

######  Theorem 5.3.

The following holds true:

  1. (1)

$\mathcal{Q}_{\theta}\in\text{Lat}C_{\varphi}$ for all inner function $\theta$
if and only if $\varphi$ is the identity map.

  2. (2)

$\mathcal{Q}_{\theta}\in\text{Lat}C_{\varphi}$ for all self-map $\varphi$ if
and only if $\theta(z)=\alpha z$ or $\theta\equiv\alpha$, where $\alpha$ is an
unimodular constant.

Proof. If $\mathcal{Q}_{\theta}\in\text{Lat}C_{\varphi}$ for all inner
function $\theta$, then, in particular,
$\mathcal{Q}_{b_{\alpha}},\mathcal{Q}_{b_{\beta}}\in\text{Lat}C_{\varphi}$,
where $\alpha\neq\beta$ are in $\mathbb{D}\setminus\\{0\\}$. By Theorem 3.1,
there exist nonzero scalars $a$ and $b$ such that

$\varphi(z)=\frac{1-a}{\bar{\alpha}}+az=\frac{1-b}{\bar{\beta}}+bz\qquad(z\in\mathbb{D}).$

Thus, comparing both sides of the above and noting the fact that
$\alpha\neq\beta$, we have $a=b=1$, which implies that $\varphi(z)=z$,
$z\in\mathbb{D}$. The converse part of $(1)$ is trivial.

Now we turn to prove $(2)$. If $\theta(z)=\alpha z$ for some unimodular
$\alpha$, then $\mathcal{Q}_{\theta}=k(\cdot,0)\mathbb{C}$ is the space of all
constant functions, which is clearly invariant under all composition operators
(cf. Remark 3.2). In the case of $\theta\equiv\alpha$, we have
$\mathcal{Q}_{\theta}=\\{0\\}$, which is also invariant under all composition
operators.

For the converse part, we first note that if $\mathcal{Q}_{\theta}=\\{0\\}$,
then $\theta H^{2}(\mathbb{D})=H^{2}(\mathbb{D})$, and hence $\theta$ is an
unimodular constant. So suppose $\mathcal{Q}_{\theta}\neq\\{0\\}$. Then there
exist $\alpha\in\mathbb{D}$ and $f\in\mathcal{Q}_{\theta}$ such that
$f(\alpha)\neq 0$. By taking $\varphi\equiv\alpha$, we see that
$1\in\mathcal{Q}_{\theta}$, which implies

$\theta(0)=\langle\theta,1\rangle=0.$

Then $\omega=\frac{\theta}{z}$ is an inner function. By Theorem 4.5, $\omega
H^{2}(\mathbb{D})$ invariant under $C_{\sigma}$ for every linear fractional
self-map $\sigma$ of $\mathbb{D}$.

Now we claim that $\omega$ has no zeros in $\mathbb{D}$. Suppose towards a
contradiction that $\omega(\alpha)=0$ for some $\alpha\in\mathbb{D}$. Since
$\omega$ is an inner function, $\omega(\beta)\neq 0$ for some
$\beta\in\mathbb{D}$. Set $\sigma=b_{\beta}\circ b_{\alpha}$ so that
$\sigma(\alpha)=\beta$. Then

$\frac{\omega\circ\sigma}{\omega}\notin H^{\infty}(\mathbb{D}),$

and hence $\omega H^{2}(\mathbb{D})$ is not invariant under $C_{\sigma}$. This
is a contradiction, and hence $\omega(z)\neq 0$, $z\in\mathbb{D}$.

Finally, consider a non-automorphic and holomorphic self-map $\sigma$ of
$\mathbb{D}$ with a fixed point in $\mathbb{D}$ (for instance, consider
$\sigma(z)=\frac{1}{2}z$). As $\omega H^{2}(\mathbb{D})$ invariant under
$C_{\sigma}$, by [2, Corollary 3.5], it follows that $\omega$ is an unimodular
constant. This implies that $\theta(z)=\alpha z$ for some unimodular constant
$\alpha$. ∎

We conclude this section by proving a reducing property of composition
operators. Here we will be restricted to finite-dimensional model spaces, as
in Theorem 3.1. Before we begin this discussion, we recall the following
result [2, Corollary 2.4]:

######  Theorem 5.4.

Let $B$ be a Blaschke product and let $\varphi$ be a holomorphic self-map of
$\mathbb{D}$. Then $BH^{2}(\mathbb{D})\in\text{Lat}C_{\varphi}$ if and only if
$\text{multiplicity}_{w}B\leq\text{multiplicity}_{w}{B\circ\varphi}$ for all
$w\in\mathbb{D}$ such that $B(w)=0$.

We are now ready for the reducing subspace property of elementary composition
operators.

######  Theorem 5.5.

Let $\varphi$ be a holomorphic self-map of $\mathbb{D}$,
$\alpha\in\mathbb{D}$, $n\geq 1$, and suppose
$\theta(z)=\left(\frac{z-\alpha}{1-\bar{\alpha}z}\right)^{n}$. Then
$\mathcal{Q}_{\theta}$ reduces $C_{\varphi}$ if and only if there exist a
scalar $c$, $|c|\leq 1$, and $\psi\in\mathcal{S}(\mathbb{D})$ such that

$\varphi(z)=\begin{cases}z\psi&\mbox{if }\alpha=0\text{ and }n=1\\\
cz&\mbox{if }\alpha=0\text{ and }n\geq 2\\\ z&\mbox{if }\alpha\neq
0.\end{cases}$

Proof. Suppose $\alpha=0$. If $n=1$, then the one-dimensional subspace
$(zH^{2}(\mathbb{D}))^{\bot}=\mathcal{Q}_{z}$ is invariant under every
composition operator (see Remark 3.2), where $zH^{2}(\mathbb{D})$ is invariant
under $C_{\varphi}$ if and only if $\varphi=z\psi$ for some
$\psi\in{\mathcal{S}}(\mathbb{D})$ (see Theorem 5.4). If $n\geq 2$, then, by
Theorem 3.1, $\mathcal{Q}_{\theta}$ is invariant under $C_{\varphi}$ if and
only if $\varphi$ is an affine map. By Theorem 5.4, $\theta H^{2}(\mathbb{D})$
is invariant under $C_{\varphi}$ if and only if $\varphi=z\psi$ for some
$\psi\in{\mathcal{S}}(\mathbb{D})$. In particular, $\varphi(0)=0$, and hence
$\theta H^{2}(\mathbb{D})$ reduces $C_{\varphi}$ if and only if
$\varphi(z)=cz$ for some $|c|\leq 1$.

Finally, assume that $\alpha\neq 0$. If $\varphi$ is the identity map, then
$C_{\varphi}=I_{H^{2}(\mathbb{D})}$, and hence $\mathcal{Q}_{\theta}$ reduces
$C_{\varphi}$. Next, we assume that $\mathcal{Q}_{\theta}$ reduces
$C_{\varphi}$. By Theorem 3.1, there exists a scalar $c(\neq 0)$ such that
$\varphi(z)=\frac{1-c}{\bar{\alpha}}+cz$. Again, by Theorem 5.4,
$(\theta\circ\varphi)(\alpha)=0$, and hence $\varphi(\alpha)=\alpha$. Now

$\frac{1-c}{\bar{\alpha}}+c\alpha=\alpha,$

is equivalent to

$(1-c)(1-|\alpha|^{2})=0.$

As $|\alpha|<1$, we must have $c=1$, that is, $\varphi(z)=z$,
$z\in\mathbb{D}$, which completes the proof. ∎

Finally, we would like to point out that the present theory of model spaces
and composition operators complement the recent paper [2]. The classification
of Beurling type invariant subspaces of composition operators ([2, Theorem
2.3] or Theorem 2.1) also plays an important role in our consideration, and
evidently, some of our results are in duality (cf. Theorem 4.3 and 4.5). On
the other hand, model spaces are fairly complex objects in operator theory and
function theory, which also represent a large class of bounded linear
operators. The present analysis clearly shows a challenging role of model
spaces to the theory of composition operators (and vice versa).

### Acknowledgments

The first author thanks the National Board for Higher Mathematics (NBHM),
India, for providing financial support to carry out this research. The second
author is supported in part by NBHM (National Board of Higher Mathematics,
India) grant NBHM/R.P.64/2014, and the Mathematical Research Impact Centric
Support (MATRICS) grant, File No : MTR/2017/000522, by the Science and
Engineering Research Board (SERB), Department of Science & Technology (DST),
Government of India.

## References

  * [1] A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81 (1948), 239-255.
  * [2] S. Bose, P. Muthukumar and J. Sarkar, Beurling type invariant subspaces of composition operators, J. Operator Theory, (2021), 14 pages. arXiv:2004.00264
  * [3] P. Bourdon, E. Fry, C. Hammond and C. Spofford, Norms of linear-fractional composition operators, Trans. Amer. Math. Soc. 356 (2004), 2459-2480.
  * [4] I. Chalendar and J. Partington, On the structure of invariant subspaces for isometric composition operators on $H^{2}(\mathbb{D})$ and $H^{2}(\mathbb{C}^{+})$, Arch. Math. (Basel). 81 (2003), 193-207.
  * [5] J. Clifford and D. Zheng, Composition operators on the Hardy space, Indiana Univ. Math. J. 48 (1999), 1585-1616.
  * [6] C. Cowen, Linear fractional composition operators on $H^{2}(\mathbb{D})$, Integral Equations and Operator Theory. 11 (1988), 151-160.
  * [7] J. Deddens, Analytic Toeplitz and composition operators, Canadian J. Math. 24 (1972), 859-865.
  * [8] R. G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413-415.
  * [9] P. L. Duren, Theory of $H^{p}$ spaces, Academic Press, New York and London, 1970.
  * [10] E. Gallardo-Gutiérrez and A. Montes-Rodríguez, The role of the angle in supercyclic behavior, J. Funct. Anal. 203 (2003), 27-43.
  * [11] E. Gallardo-Gutiérrez and R. Schroderus, The spectra of linear fractional composition operators on weighted Dirichlet spaces, J. Funct. Anal. 271 (2016), 720-745.
  * [12] S. R. Garcia, J. Mashreghi, W. T. Ross, Introduction to model spaces and their operators, Cambridge Studies in Advanced Mathematics, vol. 148. Cambridge University Press, Cambridge, 2016.
  * [13] J. Guyker, On reducing subspaces of composition operators, Acta Sci. Math. 53 (1989), 369-376.
  * [14] M. Jones, Shift invariant subspaces of composition operators on $H^{p}$, Arch. Math. (Basel) 84 (2005), 258-267.
  * [15] P. Lefèvre, D. Li, H. Queffélec and L. Rodríguez-Piazza, Some revisited results about composition operators on Hardy spaces, Rev. Mat. Iberoam. 28 (2012), 57-76.
  * [16] A. Mahvidi, Invariant subspaces of composition operators, J. Operator Theory. 46 (2001), 453-476.
  * [17] V. Matache, Invariant subspaces of composition operators, J. Operator Theory. 73 (2015), 243-264.
  * [18] A. Montes-Rodríguez, M. Ponce-Escudero and S. Shkarin, Invariant subspaces of parabolic self-maps in the Hardy space, Math. Res. Lett. 17 (2010), 99-107.
  * [19] E. Nordgren, P. Rosenthal and F. Wintrobe, Composition operators and the invariant subspace problem, C. R. Math. Rep. Acad. Sci. Canada. 6 (1984), 279-283.
  * [20] E. Nordgren, P. Rosenthal and F. Wintrobe, Invertible composition operators on $H^{p}$, J. Funct. Anal. 73 (1987), 324-344.
  * [21] H. Radjavi and P. Rosenthal, Invariant subspaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77. Springer-Verlag, New York-Heidelberg, 1973.
  * [22] J. H. Shapiro, Composition operators and classical function theory, Springer, New York, 1993.
  * [23] J. H. Shapiro, Composition operators $\heartsuit$ Toeplitz operators, Five lectures in complex analysis, 117-139, Contemp. Math., 525, Amer. Math. Soc., Providence, RI, 2010.


[a,b]Bohdan Dudar

# Development of time-of-flight particle identification for future Higgs
factories

Ulrich Einhaus  Jenny List  Konrad Helms  Frank Gaede

###### Abstract

With the emergence of advanced Silicon (Si) sensor technologies such as LGADs,
it is now possible to achieve exceptional time measurement precision below 50
ps. As a result, the implementation of time-of-flight (TOF) particle
identification (PID) for charged hadrons at future $e^{+}e^{-}$ Higgs factory
detectors has gained increasing attention. Other PID techniques require a
gaseous tracker with excellent dE/dx resolution, or a Ring-imaging Cherenkov
detector (RICH), which adds additional material in front of the calorimeter.
TOF measurements can be implemented either in the outer layers of the tracker
or in the electromagnetic calorimeter, and are thus particularly interesting
as a PID method for detector concepts based on all-silicon trackers and
optimised for particle-flow reconstruction. In this study, we will explore
potential integration scenarios of a TOF measurement in a future Higgs factory
detector, using the International Large Detector (ILD) as an example. We will
focus on the challenges associated with crucial components of TOF PID, namely
track length reconstruction and TOF measurements. The subsequent discussion
will highlight the vital impact of precise track length reconstruction and
various TOF measurement techniques, including recently developed machine
learning approaches. We will evaluate the performance in terms of $\pi/K$ and
$K/p$ separation as a function of momentum, and discuss potential physics
applications.

##  1 Introduction

The particle physics community is in an active discussion about the next
collider facility, an $e^{+}e^{-}$ collider for precision studies of the Higgs
boson, the top quark and the electroweak gauge bosons, as recommended by the
current European strategy for particle physics from 2020 and the recent
Snowmass community study. Alongside the effort to converge on an optimal
collider proposal, there is ongoing development of different detector designs
for such future Higgs factories. One open question for such detectors is the
implementation of fast $\mathcal{O}(\qty{10}{})$ timing into the detector
design. Fast timing is interesting for several aspects: time-of-flight (TOF)
particle identification (PID), 4D tracking, 5D calorimetry, background
rejection and long-lived particle searches. This study focuses on the TOF PID
of $\pi$, $K$ and $p$ using the International Large Detector (ILD) [1] as an
example case. We present the latest developments of the reconstruction
algorithms necessary for TOF PID, discuss involved challenges and present
their performance. Additionally, we briefly touch on implications for detector
concepts with different main tracker technologies and potential physics
applications of the TOF PID.

##  2 Basics of time-of-flight particle identification

For a simple helix track, the mass $m$ of a particle can be expressed in terms
of its momentum $p$, its TOF $T$ and its track length $L$, as shown in Eq. 1:

$m=p\sqrt{\frac{c^{2}T^{2}}{L^{2}}-1}\text{.}$ (1)

In a realistic detector, energy loss and scattering lead to distortions from a
simple helix, and the relation becomes more complicated [2]. However, Eq. 1
remains a good approximation if appropriate definitions of $p$ and, more
importantly, $L$ are used. Instead of the momentum of the particle at some
specific point, we use the weighted harmonic mean, as defined in Eq. 2:

$\small p=\sqrt{\langle
p^{2}\rangle_{HM}}=\sqrt{\sum^{n}_{i=1}L_{i}\bigg{/}\sum^{n}_{i=1}\frac{L_{i}}{p^{2}_{i}}}\text{.}$
(2)

The sum runs over all $n$ track segments, with $L_{i}$ being $i\text{th}$
segment length and $p_{i}$ being the momentum of the particle reconstructed at
the beginning of the given $i\text{th}$ track segment. In practice, using the
harmonic mean instead of the momentum of the particle at the IP has a
negligible impact.

##  3 Track length reconstruction

Historically TOF PID was usually limited by the TOF resolution. However, at
future Higgs factories with $\mathcal{O}(\qty{10}{})$ time resolutions, the
track length reconstruction can become a leading uncertainty factor which
limits the PID capabilities. In our study, the reconstruction of the track
length from IP to the ECAL is based on the track parametrization chosen in
ILD, which is based on five parameters: curvature $\Omega$, helix dip angle
$\tan{\lambda}$, initial azimuthal angle $\varphi_{0}$ and impact parameters
$d_{0}$ and $z_{0}$ given with respect to a reference point [3]. For a long
time, the ILD reconstruction calculated the track length using Eq. 3:

$L=\bigg{|}\frac{\varphi_{\text{ECAL}}-\varphi_{\text{IP}}}{\Omega_{\text{IP}}}\bigg{|}\sqrt{1+\tan^{2}{\lambda_{\text{IP}}}}\text{.}$
(3)

In this study, we introduce an improved method to reconstruct the track
length, defined in Eq. 4.

$L=\sum_{i=1}^{n}L_{i}=\sum_{i=1}^{n}\frac{|z_{i+1}-z_{i}|}{|\tan{\lambda}_{i}|}\sqrt{1+\tan^{2}{\lambda}_{i}}$
(4)

Both Eqs. 3 and 4 are equivalent for any perfectly helical track. However, on
the fully reconstructed particles from the full ILD simulation Monte-Carlo
(MC) samples, one can see a significant improvement of the mass reconstruction
in Fig. 1. This is due to the fact that fully reconstructed tracks are not
exactly perfect helices and experience numerous effects such as energy loss as
well as track fitting uncertainties.

(a) Old track length (Eq. 3) used in [1].

(b) New track length (Eq. 4).

Figure 1: The mass of charged hadrons reconstructed from TOF as a function of
their momentum for previous and improved track length reconstruction. In both
cases, perfect time information is assumed for illustration.

Figure 1 shows that a new approach using Eq. 4 results in significantly
better-reconstructed masses compared to the previous approach using Eq. 3. The
improvement comes from three major factors. Firstly, Eq. 3 reconstructs a
helix length based on only two points, which does not work for tracks with
multiple curls, as expected for low momentum particles. Secondly, it has been
observed that using $\tan{\lambda}$ and $z$ instead of $\Omega$ and $\varphi$
as base parameters of the track evolution results in better performance since
the former tends to be reconstructed with better precision than the latter.
Lastly, Eq. 4 uses multiple hits along the track summing over the lengths of
many small helix pieces, which are built from the local helix track parameters
at every tracker hit utilizing the full information of the Kalman filter, that
is used in the track fit.This accounts for energy loss effects along the
track.

##  4 Time-of-flight reconstruction

A dedicated timing layer provides robust TOF measurement, but its
effectiveness is constrained by the single measurement time resolution.
Integration of time measurements into the electromagnetic calorimeter (ECAL)
potentially improves TOF resolution as $1/\sqrt{N}$, with $N$ being the number
of ECAL hits with time information.

In this section, we study TOF reconstruction using the ILD ECAL, a highly
granular Si/W sandwich calorimeter with thirty layers of $5\times
5\,$\mathrm{m}\mathrm{m}^{2}$$ cells [1]. We consider three scenarios for the
different time measurement implementation scenarios in ILD assuming different
hit time resolutions: 1) using only the first ECAL layer with a 30 time
resolution; 2) utilising the double strip layer of ILD Silicon External
Tracker (SET) with a 50 hit time resolution; 3) using hits in the first ten
ECAL layers, selecting in each layer only the closest hit to the track’s
extrapolation line in the ECAL with 100 hit time resolution.

In all cases, we correct the ECAL hits’ time information by their distance
from the track’s extrapolated entry point into the ECAL, divided by the speed
of light. The reconstructed TOF is the average of corrected time measurements.
We compare this to the MC true TOF information, which is defined as the
corrected MC truth time of the closest hit to the track’s extrapolated entry
point into the ECAL. The performance of these methods is presented in Fig. 2.

(a) Linear scale.

(b) Log scale.

Figure 2: TOF resolution obtained using three different hypothetical scenarios
of time measurement integration in ILD using a single ECAL layer, double SET
layer or ten ECAL layers with different hit time resolutions.

The third method, highlighted in green, shows that it is possible to use
multiple hits from in-depth of the ECAL shower to get $\approx\qty{33.7}{}$
TOF resolution using 100 hit time resolution if one does a hit preselection to
ensure that hits further away from the track don’t introduce a significant
bias. Despite this preselection, the third method still has distinctive non-
Gaussian tails, visible in Fig. 2. Using a machine learning (ML) approach for
the TOF reconstruction could provide significant improvements over the
conventional method by using the full information to its utmost potential. We
develop an ML model based on the EPiC encoder [4] followed by a multi-layer
perceptron using all spatial, time and energy information of the ECAL hits as
well as reconstructed track parameters, such as momentum, length and
extrapolated entry point into the ECAL. Though the ML model is still under
development, the first results look already promising.

##  5 Separation power for time-of-flight particle identification

The PID performance has been studied for various TOF resolutions with the
latest track length reconstruction. The separation power between particle
hypotheses is evaluated following the method of [5]: The overlap between the
reconstructed mass distributions for pions and kaons is calculated. Then two
normal distributions with $\mu_{1}$, $\mu_{2}$ and
$\sigma:=\sigma_{1}=\sigma_{2}$ are placed at a distance
$\lvert\mu_{1}-\mu_{2}\rvert$ with the same overlap value, leading to a well-
defined separation power $S=\lvert\mu_{1}-\mu_{2}\rvert/\sigma$.

(a) $\pi$/$K$ separation.

(b) $K$/$p$ separation.

Figure 3: The separation power between charged hadrons particles based on TOF
PID, with the latest track length reconstruction and different TOF resolutions
as a function of the momentum.

Figure 3 shows that with perfect TOF resolution, only limited by the track
length reconstruction, one could achieve $\pi$/$K$ ($K$/$p$) separation at the
3$\sigma$ level for momenta up to 8 (17). This shows a significant potential
if the TOF resolution per particle could be improved beyond 10, e.g. by ML
means utilising multiple independent time measurements from the ECAL, as
discussed in the previous section. In the case of a single time measurement
with a time resolution of 50 the momentum reach of TOF PID becomes
significantly limited reaching $\pi$/$K$ ($K$/$p$) separation at the 3$\sigma$
level up to 2.1 (3.7).

##  6 Potential physics applications

Generally, kaon-ID is a crucial tool for the precision measurements planned at
a future collider facility. TOF PID works on a limited low-momentum range and
nicely complements other PID tools such as dE/dx, which perform best at high
momentum and have a notable ’blind spot’ at low momentum. The most common
processes at $E_{\text{CM}}=\qty{250}{}$ $Z\rightarrow q\bar{q}$ and
$WW\rightarrow q\bar{q}q\bar{q}$ contain a sizable fraction of charged hadrons
at low momentum, which is covered by TOF easily. For illustration, about 38%
(51%) of the $K$ ($p$) have a momentum below 3, which should be identifiable
with TOF assuming 30 TOF resolution, which is deemed feasible with state-of-
the-art technology with a single time measurement. Enhanced PID capabilities
of a detector improved by TOF PID have the potential to contribute to various
precision studies in flavour physics, Higgs sector, $q/\bar{q}$ separation for
$A_{\mathrm{FB}}$, $V_{\mathrm{cs}}$ and $V_{\mathrm{cb}}$ measurements, as
well as to enhance event reconstruction, extend long-lived particle searches
and measure the charged kaon mass. A quantitative assessment of TOF PID
contributions for various physics studies is a subject of further studies.

##  7 Summary

We introduced a novel track length reconstruction method, highlighting its
importance for mass reconstruction. Our study presents the results for the ML-
based TOF estimator, using time information in the first ten ECAL layers with
a 50 hit resolution. This technique yields a TOF resolution at the
distribution core $\approx\qty{6.7}{}$, outperforming the previous state-of-
the-art analytical algorithm. We studied the impact of TOF resolution on the
momentum reach for particle separation using our novel track length estimator.
A TOF resolution per particle better than 30 significantly improves the
momentum reach, particularly as it approaches 0. A resolution worse than 30
provides limited momentum reach, diminishing as the resolution worsens. TOF
could nicely complement a PID system of a detector at a future Higgs factory
and potentially enhance precision studies reliant on PID. However, a
quantitative assessment of the benefits requires future studies.

##  8 Acknowledgements

We would like to thank the LCC generator working group and the ILD software
working group for providing the simulation and reconstruction tools and
producing the Monte Carlo samples used in this study. This work has benefited
from computing services provided by the ILC Virtual Organisation, supported by
the national resource providers of the EGI Federation and the Open Science
GRID. In this study we widely used the National Analysis Facility (NAF) [6]
and would like to thank Grid computational resources operated at Deutsches
Elektronen-Synchrotron (DESY), Hamburg, Germany. We thankfully acknowledge the
support by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy EXC 2121 "Quantum Universe"
390833306.

## References

  * [1] The ILD Collaboration, _International Large Detector: Interim Design Report_ , 2020.  [physics.ins-det/2003.01116].
  * [2] W.A. Mitaroff, _Time-of-flight estimation by utilizing Kalman filter tracking information – Part I: the concept_ , 2021.  [physics.ins-det/2107.02031].
  * [3] T. Kraemer and DESY, _Track Parameters in LCIO_ , _LC-Notes (2006)._ (2006) .
  * [4] E. Buhmann, G. Kasieczka and J. Thaler, _EPiC-GAN: Equivariant Point Cloud Generation for Particle Jets_ , 2023.  [hep-ph/2301.08128].
  * [5] K. Götzen, _PANDA Quality Measures for PID Classification Problems_ , 2017\.  Panda PID Computing Workshop [https://indico.gsi.de/event/7080/contributions/31950/].
  * [6] A. Haupt and Y. Kemp, _The NAF: National analysis facility at DESY_ , _Journal of Physics: Conference Series_ 219 (2010) 052007.