# Summability and duality

Soumitra Ghara  Département de mathématiques et de statistique, Université
Laval, Québec City (Québec), Canada G1V 0A6<EMAIL_ADDRESS>,  Javad
Mashreghi  Département de mathématiques et de statistique, Université Laval,
Québec City (Québec), Canada G1V 0A6<EMAIL_ADDRESS>and
Thomas Ransford  Département de mathématiques et de statistique, Université
Laval, Québec City (Québec), Canada G1V 0A6<EMAIL_ADDRESS>
###### Abstract.

We formalize the observation that the same summability methods converge in a
Banach space $X$ and its dual $X^{*}$. At the same time we determine
conditions under which these methods converge in the weak and weak*-topologies
on $X$ and $X^{*}$ respectively. We also derive a general limitation theorem,
which yields a necessary condition for the convergence of a summability method
in $X$. These results are then illustrated by applications to a wide variety
of function spaces, including spaces of continuous functions, Lebesgue spaces,
the disk algebra, Hardy and Bergman spaces, the BMOA space, the Bloch space,
and de Branges–Rovnyak spaces. Our approach shows that all these applications
flow from just two abstract theorems.

###### Key words and phrases:

Summability, Limitation theorem, Cesàro mean, Banach space, Dual space

###### 2020 Mathematics Subject Classification:

Primary 46A35, Secondary 30H10, 30H20, 30H45

Ghara supported by a Fields–Laval postdoctoral fellowship. Mashreghi supported
by an NSERC Discovery Grant. Ransford supported by grants from NSERC and the
Canada Research Chairs program.

##  1\. Introduction

Let $X$ be a Banach space of holomorphic functions on the open unit disk
${\mathbb{D}}$, and suppose that $X$ contains the polynomials. Every function
$f\in X$ has a Taylor expansion $f(z)=\sum_{j\geq 0}a_{j}z^{j}$, which
converges locally uniformly on ${\mathbb{D}}$ to $f(z)$. However, it can
happen that the series fails to converge to $f$ in the norm of $X$. This is
the case, for example, whenever polynomials are not dense in $X$, but it may
occur even when they are dense. Here is a short list of examples illustrating
various possibilities, ranging from ‘best’ to ‘worst’.

  1. (1)

If $X$ is the Hardy space $H^{2}$, then the Taylor series of $f$ does converge
to $f$ in the norm of $X$. The same is true if $X$ is the Dirichlet space or
the Bergman space.

  2. (2)

If $X$ is the disk algebra $A({\mathbb{D}})$, then the Taylor series of $f$
may fail to converge in the norm of $X$ (du Bois-Reymond’s example), but its
Cesàro means do converge in norm (Fejér’s theorem).

  3. (3)

If $X$ is a de Branges–Rovnyak space ${\mathcal{H}}(b)$, then, for certain
choices of $b$ and $f$, the Cesàro means may fail to converge in norm, though
polynomials are still dense in $X$ (see [3]).

  4. (4)

If $X$ fails to have the bounded approximation property, then no lower-
triangular summation method can converge in norm for every function, even
though polynomials may still be dense in $X$ (see [13]).

In this article we are mainly interested in cases like (2) and (3), where some
summability methods work and others do not, and the problem is to determine
the range of methods that do work. Our starting point is the fact that,
typically, the same methods tend to work in $X$ and in its dual $X^{*}$. This
is because the convergence of a summability method often boils down to whether
a certain sequence of summation operators is uniformly bounded in norm, and an
operator has the same norm as its adjoint. We formalize this idea, at the same
time linking it to weak- and weak*-convergence in $X$ and $X^{*}$
respectively. We also derive a general limitation theorem, namely a necessary
condition for the convergence of summability method in a given Banach space.
The proofs of these results are carried out in two steps: in §2 we establish a
general operator-theoretic result, which is then used in §3 to derive the
abstract summability theorems.

The rest of the article is devoted to various examples and applications of
these results. In §4 we consider continuous-function and Lebesgue spaces, as
well as the disk algebra. In §5 we treat the Hardy and Bergman spaces and
their relatives, $\operatorname{BMOA}$ and the Bloch space. Finally, in §6, we
consider reproducing kernel Hilbert spaces of holomorphic functions, and, in
particular, de Branges–Rovnyak spaces. Some of the applications are already
known, others are slight generalizations of known results, and some are
completely new. Our approach shows that they all flow from just two abstract
theorems.

##  2\. Operator theory

In what follows, $X$ is a real or complex Banach space. We write $X^{*}$ for
the dual space of $X$, and $\langle\cdot,\cdot\rangle$ for the duality pairing
between $X$ and $X^{*}$. We use $w$ and $w^{*}$ to denote the weak and weak*
topologies on $X$ and $X^{*}$ respectively. Lastly, given a bounded operator
$T$ on $X$, we write $T^{*}$ for the adjoint operator on $X^{*}$, defined by
the relation

$\langle x,T^{*}\phi\rangle=\langle Tx,\phi\rangle\quad(x\in X,~{}\phi\in
X^{*}).$

The purpose of this section is to establish the following result.

######  Theorem 2.1.

Let $(T_{n})_{n\geq 0}$ be a sequence of bounded, finite-rank operators on $X$
such that

(2.1) $T_{n}T_{m}(X)\subset T_{m}(X)\quad\text{and}\quad
T_{n}^{*}T_{m}^{*}(X^{*})\subset T_{m}^{*}(X^{*})\qquad(m,n\geq 0).$

Let

$Y:=\overline{\operatorname{span}(\cup_{m\geq 0}T_{m}(X))}\quad\text{and}\quad
Z:=\overline{\operatorname{span}(\cup_{m\geq 0}T_{m}^{*}(X^{*}))},$

where the closures are taken in the norm topologies of $X$ and $X^{*}$
respectively. Then the following statements are equivalent:

  1. cmrmn(i)

$T_{n}x\to x$ in $(X,w)$ for all $x\in X$;

  2. cmrmn(ii)

$T_{n}x\to x$ in $(X,\|\cdot\|)$ for all $x\in X$;

  3. cmrmn(iii)

$T_{n}^{*}\phi\to\phi$ in $(X^{*},w^{*})$ for all $\phi\in X^{*}$;

  4. cmrmn(iv)

$T_{n}^{*}\phi\to\phi$ in $(X^{*},w^{*})$ for all $\phi\in Z$, and $Z$ is
$w^{*}$-sequentially dense in $X^{*}$ and $Y=X$;

  5. cmrmn(v)

$T_{n}^{*}\phi\to\phi$ in $(X^{*},\|\cdot\|)$ for all $\phi\in Z$, and $Z$ is
$w^{*}$-sequentially dense in $X^{*}$ and $Y=X$.

If, further, $X$ is reflexive, then these are equivalent to:

  1. cmrmn(vi)

$T_{n}^{*}\phi\to\phi$ in $(X^{*},\|\cdot\|)$ for all $\phi\in X^{*}$.

We shall prove this result via a series of lemmas, beginning with a very
simple one.

######  Lemma 2.2.

Let $(T_{n})_{n\geq 0}$ be a sequence of bounded operators on $X$. The
following statements are equivalent:

  1. cmrmn(i)

$T_{n}x\to x$ in $(X,w)$ for all $x\in X$;

  2. cmrmn(ii)

$T_{n}^{*}\phi\to\phi$ in $(X^{*},w^{*})$ for all $\phi\in X^{*}$.

###### Proof.

We have

$\displaystyle T_{n}x\overset{w}{\longrightarrow}x\quad\forall x\in X$
$\displaystyle\iff\langle T_{n}x,\phi\rangle\to\langle
x,\phi\rangle\quad\forall\phi\in X^{*},\forall x\in X$
$\displaystyle\iff\langle x,T_{n}^{*}\phi\rangle\to\langle
x,\phi\rangle\quad\forall x\in X,\forall\phi\in X^{*}$ $\displaystyle\iff
T_{n}^{*}\phi\overset{w^{*}}{\longrightarrow}\phi\quad\forall\phi\in
X^{*}.\qed$

We next establish a similar result relating weak convergence in $X$ to norm
convergence. In order to obtain an equivalence, we need to impose some
conditions on the operators $T_{n}$.

######  Lemma 2.3.

Let $(T_{n})_{n\geq 0}$ be a sequence of bounded, finite-rank operators on $X$
such that

(2.2) $T_{n}T_{m}(X)\subset T_{m}(X)\qquad(m,n\geq 0).$

Then the following statements are equivalent:

  1. cmrmn(i)

$T_{n}x\to x$ in $(X,w)$ for all $x\in X$;

  2. cmrmn(ii)

$T_{n}x\to x$ in $(X,\|\cdot\|)$ for all $x\in X$.

######  Remarks.

(1) Lemma 2.3 fails without the assumption ‘finite-rank’. For example, if
$X:=\ell^{2}({\mathbb{Z}}^{+})$ and $T_{n}:=I+S^{n}$, where $S$ is the
unilateral shift on $\ell^{2}({\mathbb{Z}}^{+})$, then $T_{n}T_{m}=T_{m}T_{n}$
for all $m,n$, so (2.2) holds, and

$T_{n}x-x=S^{n}x\overset{w}{\longrightarrow}0\quad\forall
x\in\ell^{2}({\mathbb{Z}}^{+}),$

but

$\|T_{n}e_{0}-e_{0}\|_{2}=\|e_{n}\|_{2}\not\to 0.$

(2) Lemma 2.3 also fails without the assumption (2.2). For example, if
$X=\ell^{2}({\mathbb{Z}}^{+})$ and $T_{n}:=\sum_{j=0}^{n}(e_{j}\otimes
e_{j})+(e_{n}\otimes e_{0})$, then each $T_{n}$ is a bounded, finite-rank
operator, and

$T_{n}x-x=-\sum_{j>n}\langle x,e_{j}\rangle e_{j}+\langle x,e_{0}\rangle
e_{n}\overset{w}{\longrightarrow}0\quad\forall
x\in\ell^{2}({\mathbb{Z}}^{+}),$

but

$\|T_{n}e_{0}-e_{0}\|_{2}=\|e_{n}\|_{2}\not\to 0.$

Note that, in this example, if $m<n$, then
$T_{m}(X)=\operatorname{span}\\{e_{0},e_{1},\dots,e_{m}\\}$, while
$T_{n}T_{m}(X)=\operatorname{span}\\{e_{0}+e_{n},e_{1},\dots,e_{m}\\}$.

###### Proof of Lemma 2.3.

It is enough to prove that (i) $\Rightarrow$ (ii), since the reverse
implication is obvious. Suppose then that $T_{n}x\to x$ weakly for all $x\in
X$. We need to show that it also converges in norm. This will be done in four
steps.

The first step is to show that $\|T_{n}x-x\|\to 0$ if
$x\in\operatorname{span}(\cup_{m}T_{m}(X))$. Fix $m$ and let $\psi$ be a
continuous linear functional on $T_{m}(X)$. By the Hahn–Banach theorem, we can
extend $\psi$ to a continuous linear functional $\phi$ on the whole of $X$.
Therefore, for all $x\in T_{m}(X)$, we have

$\langle T_{n}x,\psi\rangle=\langle T_{n}x,\phi\rangle\to\langle
x,\phi\rangle=\langle x,\psi\rangle.$

This shows that $T_{n}x\to x$ in the weakly in $T_{m}(X)$. As
$\dim(T_{m}(X))<\infty$, the weak topology and norm topology coincide, so
$\|T_{n}x-x\|\to 0$ for all $x\in T_{m}(X)$. Finally, by linearity, it follows
that $\|T_{n}x-x\|\to 0$ for all $x\in\operatorname{span}(\cup_{m}T_{m}(X))$,
as claimed.

The second step is to show that $\operatorname{span}(\cup_{m}T_{m}(X))$ is
norm-dense in $X$. Suppose the contrary. Then, by the Hahn–Banach theorem,
there exists $\phi\in X^{*}\setminus\\{0\\}$ such that $\phi=0$ on
$\operatorname{span}(\cup_{m}T_{m}(X))$. For all $m\geq 1$, we have

$\langle x,T_{m}^{*}\phi\rangle=\langle T_{m}x,\phi\rangle=0\quad(x\in X),$

so $T_{m}^{*}\phi=0$. Also, from Lemma 2.2, we know that
$T_{m}^{*}\phi\to\phi$ in $(X^{*},w^{*})$. Hence $\phi=0$. This contradicts
the choice of $\phi$. We conclude that, as claimed,
$\operatorname{span}(\cup_{m}T_{m}(X))$ is norm-dense in $X$.

The third step is to show that $\sup_{n}\|T_{n}\|<\infty$. For each $x\in X$,
the sequence $(T_{n}x)$ converges weakly, so it is weakly bounded. By the
Banach–Steinhaus theorem, it is also norm-bounded, i.e.,
$\sup_{n}\|T_{n}x\|<\infty$. As this holds for each $x\in X$, a second
application of Banach–Steinhaus shows that $\sup_{n}\|T_{n}\|<\infty$, as
claimed.

The fourth and final step is to show that $\|T_{n}x-x\|\to 0$ for all $x\in
X$. Let $x\in X$ and let $\epsilon>0$. By steps 2 and 3, there exists
$x_{0}\in\operatorname{span}(\cup_{m}T_{m}(X))$ such that
$\|x-x_{0}\|<\epsilon/(1+\sup_{n}\|T_{n}\|)$. By step 1, there exists $n_{0}$
such that $\|T_{n}x_{0}-x_{0}\|<\epsilon$ for all $n\geq n_{0}$. Then, if
$n\geq n_{0}$, we have

$\|T_{n}x-x\|\leq\|T_{n}(x-x_{0})\|+\|T_{n}x_{0}-x_{0}\|+\|x_{0}-x\|<3\epsilon.$

Thus $\|T_{n}x-x\|\to 0$, as was to be proved. ∎

If $X$ is reflexive, then we may interchange the roles of $X$ and $X^{*}$ in
Lemma 2.3, and deduce the following corollary.

######  Corollary 2.4.

Suppose that $X$ is reflexive. Let $(T_{n})_{n\geq 0}$ be a sequence of
bounded, finite-rank operators on $X$ such that

(2.3) $T_{n}^{*}T_{m}^{*}(X^{*})\subset T_{m}^{*}(X^{*})\qquad(m,n\geq 0).$

Then the following statements are equivalent:

  1. cmrmn(i)

$T_{n}^{*}\phi\to\phi$ in $(X^{*},w^{*})$ for all $\phi\in X^{*}$;

  2. cmrmn(ii)

$T_{n}^{*}\phi\to\phi$ in $(X^{*},\|\cdot\|)$ for all $\phi\in X^{*}$.

######  Remark.

If $X$ is not reflexive, then Corollary 2.4 may break down. For example, let
$X:=\ell^{1}({\mathbb{Z}}^{+})$ and let
$T_{n}:\ell^{1}({\mathbb{Z}}^{+})\to\ell^{1}({\mathbb{Z}}^{+})$ be the
projection onto the first $n$ coordinates. Its adjoint
$T_{n}^{*}:\ell^{\infty}({\mathbb{Z}}^{+})\to\ell^{\infty}({\mathbb{Z}}^{+})$
is also the projection onto the first $n$ coordinates. The sequences $(T_{n})$
and $(T_{n}^{*})$ satisfy (2.2) and (2.3) respectively. Also
$\|T_{n}x-x\|_{1}\to 0$ for all $x\in\ell^{1}({\mathbb{Z}}^{+})$, so by Lemmas
2.2 and 2.3 we have $T_{n}^{*}\phi\to\phi$ weak* for all
$\phi\in\ell^{\infty}({\mathbb{Z}}^{+})$. However, if
$\phi_{0}:=(1,1,1,\dots)$, then
$\|T_{n}^{*}\phi_{0}-\phi_{0}\|_{\infty}\not\to 0$. Note that, in this
example, the norm-closure of $\operatorname{span}\cup_{n}T_{n}^{*}(X^{*})$ is
$c_{0}$.

Here is a version of Corollary 2.4 valid for all $X$, not necessarily
reflexive.

######  Lemma 2.5.

Let $(T_{n})_{n\geq 0}$ be a sequence of bounded, finite-rank operators on $X$
such that

(2.4) $T_{n}(T_{m}(X))\subset T_{m}(X)\quad\text{and}\quad
T_{n}^{*}(T_{m}^{*}(X^{*}))\subset T_{m}^{*}(X^{*})\quad(m,n\geq 0).$

Let

$Y:=\overline{\operatorname{span}(\cup_{m\geq 0}T_{m}(X))}\quad\text{and}\quad
Z:=\overline{\operatorname{span}(\cup_{m\geq 0}T_{m}^{*}(X^{*}))},$

where the closures are taken in the norm topologies of $X$ and $X^{*}$
respectively. Then the following statements are equivalent:

  1. cmrmn(i)

$T_{n}^{*}\phi\to\phi$ in $(X^{*},w^{*})$ for all $\phi\in X^{*}$;

  2. cmrmn(ii)

$T_{n}^{*}\phi\to\phi$ in $(X^{*},w^{*})$ for all $\phi\in Z$, and $Z$ is
$w^{*}$-sequentially dense in $X^{*}$ and $Y=X$;

  3. cmrmn(iii)

$T_{n}^{*}\phi\to\phi$ in $(X^{*},\|\cdot\|)$ for all $\phi\in Z$, and $Z$ is
$w^{*}$-sequentially dense in $X^{*}$ and $Y=X$.

For the proof, we need a further lemma.

######  Lemma 2.6.

Let $Z$ be a subspace of $X^{*}$ that is $w^{*}$-sequentially dense in
$X^{*}$. Then the $w^{*}$-closure of the unit ball of $Z$ contains a positive
multiple of the unit ball of $X^{*}$.

###### Proof.

Let $C$ be the closure in $(X^{*},w^{*})$ of the unit ball of $Z$. Given
$\phi\in X^{*}$, there exists a sequence $(\phi_{n})$ in $Z$ such that
$\phi_{n}$ is $w^{*}$-convergent to $\phi$. By the Banach–Steinhaus theorem,
since $(\phi_{n})$ is $w^{*}$-bounded, it is norm-bounded. Hence there exists
an integer $m\geq 1$ such that $\phi\in mC$. Thus we have $\cup_{m\geq
1}mC=X^{*}$. Each set $mC$ is $w^{*}$-closed in $X^{*}$, so it is certainly
norm-closed. We may therefore apply Baire’s theorem to deduce that there
exists $m_{0}$ such that $m_{0}C$ has non-empty norm-interior. As $m_{0}C$ is
a convex, symmetric set, it follows that $0$ belongs to the norm interior of
$m_{0}C$. In other words, $C$ contains a ball around $0$. ∎

###### Proof of Lemma 2.5.

[(i)$\Rightarrow$(ii)]: Suppose that (i) holds. Then it is obvious that
$T_{n}^{*}\phi\to\phi$ for all $\phi\in Z$, and also that $Z$ is
$w^{*}$-sequentially dense in $X^{*}$. Finally, by Lemmas 2.2 and 2.3, (i)
implies that $T_{n}x\to x$ in norm for all $x\in X$, and this entails that
$Y=X$.

[(ii)$\Rightarrow$(iii)]: Suppose that (ii) holds. By (ii), we have
$T_{n}^{*}\phi\to\phi$ in $(X^{*},w^{*})$ for each $\phi\in Z$. By the
Banach–Steinhaus theorem, applied to the sequence $(T_{n}^{*}|_{Z})$ on $Z$,
we have $\sup_{n}\|T_{n}^{*}|_{Z}\|<\infty$.

Let $m\geq 0$. Then $T_{n}^{*}\phi\to\phi$ in $(X^{*},w^{*})$ for all $\phi\in
T_{m}^{*}(X^{*})$. Since $\dim(T_{m}^{*}(X^{*}))<\infty$, it follows that
$T_{n}^{*}\phi\to\phi$ in norm for all $\phi\in T_{m}^{*}(X^{*})$. As this
holds for each $m\geq 0$, we deduce that $T_{n}^{*}\phi\to\phi$ in norm for
all $\phi\in\operatorname{span}(\cup_{m\geq 0}T_{m}^{*}(X^{*}))$. Lastly, as
$\sup_{n}\|T_{n}^{*}|_{Z}\|<\infty$, it follows that $T_{n}^{*}\phi\to\phi$ in
norm for all $\phi\in Z$.

[(iii)$\Rightarrow$(i)]: Suppose that (iii) holds. As noted above,
$\sup_{n}\|T_{n}^{*}|_{Z}\|<\infty$. By (iii) and Lemma 2.6, the unit ball of
$Z$ is $w^{*}$-dense in a ball of radius $r>0$ in $X^{*}$. It follows that,
for every operator $T$ on $X$, we have $\|T^{*}\|\leq\|T^{*}|_{Z}\|/r$.
Combining these facts, we deduce that $K:=\sup_{n}\|T_{n}^{*}\|<\infty$.

Let $m\geq 0$ and let $Z_{m}:=\\{\phi|_{T_{m}(X)}:\phi\in Z\\}$. Since $Z$ is
$w^{*}$-sequentially dense in $X^{*}$ and $\dim T_{m}(X)<\infty$, it follows
that $Z_{m}$ is norm-dense in $T_{m}(X)^{*}$. Let $x\in T_{m}(X)$. Let
$\psi\in T_{m}(X)^{*}$ and let $\epsilon>0$. Then there exists $\phi\in Z$
with $\|\psi-\phi|_{T_{m}(X)}\|<\epsilon$. Since $\|T_{n}^{*}\phi-\phi\|\to
0$, there exists $N$ such that

$n\geq N\quad\Rightarrow\quad\|T_{n}^{*}\phi-\phi\|<\epsilon.$

Then

$\langle T_{n}x-x,\,\psi\rangle=\langle x,T_{n}^{*}\phi-\phi\rangle+\langle
T_{n}x-x,\,\psi-\phi|_{T_{m}(X)}\rangle,$

so, for all $n\geq N$,

$|\langle
T_{n}x-x,\,\psi\rangle|\leq\|x\|\|T_{n}^{*}\phi-\phi\|+(K+1)\|x\|\|\psi-\phi|_{T_{m}(X)}\|\leq(\|x\|+K+1)\epsilon.$

Thus $T_{n}x\to x$ weakly for all $x\in T_{m}(X)$. As $\dim T_{m}(X)<\infty$,
it follows that $T_{n}x\to x$ in norm for all $x\in T_{m}(X)$. As this holds
for all $m\geq 0$, we deduce that $T_{n}x\to x$ in norm for all
$x\in\operatorname{span}(\cup_{m}T_{m}(X))$. Since
$\sup_{n}\|T_{n}\|=\sup_{n}\|T_{n}^{*}\|=K<\infty$, it follows that $T_{n}x\to
x$ for all $x\in Y$.

By (iii), we have $Y=X$. Thus $T_{n}x\to x$ in norm for all $x\in X$. By
Lemmas 2.2 and 2.3, this implies $T_{n}^{*}\phi\to\phi$ in $(X^{*},w^{*})$ for
all $\phi\in X^{*}$. ∎

######  Remark.

Although statements (i)–(iii) are all about the adjoint operators $T_{n}^{*}$,
we nonetheless need the invariance assumption that $T_{n}(T_{m}(X))\subset
T_{m}(X)$ in (2.4). This assumption is used in the proof of the implication
(iii)$\Rightarrow$(i), and the result is actually false without this
assumption. Here is a counterexample.

Let $X:=\ell^{1}({\mathbb{N}})$. For $n\geq 1$ define
$T_{n}:\ell^{1}({\mathbb{N}})\to\ell^{1}({\mathbb{N}})$ by

$T_{n}:=P_{2n}(S^{n}+I),$

where $S:\ell^{1}({\mathbb{N}})\to\ell^{1}({\mathbb{N}})$ is the unilateral
shift and $P_{2n}:\ell^{1}({\mathbb{N}})\to\ell^{1}({\mathbb{N}})$ is the
projection onto the first $2n$ coordinates. Taking adjoints, we have
$T_{n}^{*}:\ell^{\infty}({\mathbb{N}})\to\ell^{\infty}({\mathbb{N}})$ given by
$T_{n}^{*}=(S^{*n}+I)P_{2n}$. The following properties are easily verified:

  * •

$T_{m}(X)=\operatorname{span}\\{e_{1},\dots,e_{2m}\\}$ for all $m\geq 1$.

  * •

$T_{m}^{*}(X^{*})=\operatorname{span}\\{e_{1},\dots,e_{2m}\\}$ for all $m\geq
1$.

  * •

$T_{n}^{*}(T_{m}^{*}(X^{*}))\subset T_{m}^{*}(X^{*})$ for all $m,n\geq 1$.

  * •

$Y=\ell^{1}({\mathbb{N}})=X$.

  * •

$Z=c_{0}({\mathbb{N}})$, which is $w^{*}$-sequentially dense in
$X^{*}=\ell^{\infty}({\mathbb{N}})$.

If $\phi\in c_{0}({\mathbb{N}})$, then we have

$\|T_{n}^{*}\phi-\phi\|_{\infty}\leq\|S^{*n}\phi\|_{\infty}+\|P_{2n}\phi-\phi\|_{\infty}\to
0\quad(n\to\infty).$

Thus $T_{n}^{*}\phi\to\phi$ in norm for all $\phi\in Z$. Therefore (iii)
holds.

However, if $\phi\in\ell^{\infty}({\mathbb{N}})\setminus c_{0}({\mathbb{N}})$,
then

$\langle e_{1},\,T_{n}^{*}\phi-\phi\rangle=\langle
T_{n}e_{1}-e_{1},\,\phi\rangle=\langle e_{n+1},\phi\rangle\not\to
0\quad(n\to\infty),$

and so $T_{n}^{*}\phi\not\to\phi$ weak*. Therefore (i) fails.

Finally, the main result of the section, Theorem 2.1, follows by combining
Lemmas 2.2, 2.3 and 2.5.

##  3\. Summability

###  3.1. The basic set-up

In the rest of the paper, we consider the following set-up. As before, $X$
denotes a Banach space with dual space $X^{*}$. Let $(e_{k})_{k\geq 0}$ and
$(\psi_{k})_{k\geq 0}$ be sequences in $X$ and $X^{*}$ respectively such that

$\left\\{\begin{aligned} \langle e_{j},\psi_{k}\rangle&=0,\quad\forall
j,k,~{}j\neq k,\\\ \langle e_{k},\psi_{k}\rangle&\neq 0,\quad\forall
k.\end{aligned}\right.$

For each $k\geq 0$, define $P_{k}:X\to X$ by

$P_{k}:=\frac{e_{k}\otimes\psi_{k}}{\langle e_{k},\psi_{k}\rangle}.$

Explicitly,

$P_{k}(x):=\frac{\langle x,\psi_{k}\rangle}{\langle
e_{k},\psi_{k}\rangle}e_{k}\quad(x\in X).$

Clearly $P_{k}(e_{k})=e_{k}$ and $P_{k}(e_{j})=0$ if $j\neq k$. It is easy to
see that $P_{k}$ is a rank-one projection ($P_{k}^{2}=P_{k}$) with

$\|P_{k}\|=\frac{\|e_{k}\|\|\psi_{k}\|}{|\langle e_{k},\psi_{k}\rangle|}.$

Its adjoint $P_{k}^{*}:X^{*}\to X^{*}$ is given by

$P_{k}^{*}=\frac{\psi_{k}\otimes e_{k}}{\langle e_{k},\psi_{k}\rangle}.$

Let $A=(a_{nk})_{n,k\geq 0}$ be an infinite matrix of complex scalars such
that, for each $n\geq 0$,

(3.1) $\sum_{k\geq 0}|a_{nk}|\|P_{k}\|<\infty.$

We define the associated summation operators $S_{n}^{A}:X\to X~{}(n\geq 0)$ by
the absolutely convergent series

$S_{n}^{A}:=\sum_{k\geq 0}a_{nk}P_{k}.$

Explicitly, we have

$\displaystyle S_{n}^{A}(x)$ $\displaystyle=\sum_{k\geq 0}a_{nk}\frac{\langle
x,\psi_{k}\rangle}{\langle e_{k},\psi_{k}\rangle}e_{k}\quad(x\in X),$
$\displaystyle(S_{n}^{A})^{*}(\phi)$ $\displaystyle=\sum_{k\geq
0}a_{nk}\frac{\langle e_{k},\phi\rangle}{\langle
e_{k},\psi_{k}\rangle}\psi_{k}\quad(\phi\in X^{*}).$

###  3.2. The main equivalence

With this notation established, our goal is to prove the following theorem.

######  Theorem 3.1.

Let $Z$ be the norm-closure in $X^{*}$ of
$\operatorname{span}\\{\psi_{k}:k\geq 0\\}$. Suppose that there exists at
least one matrix $A^{0}$ satisfying (3.1) such that $S_{n}^{A^{0}}x\to x$
(either weakly or in norm) for all $x\in X$. Then, for every matrix $A$
satisfying (3.1), the following statements are equivalent:

  1. cmrmn(i)

$S_{n}^{A}(x)\to x$ in $(X,w)$ for all $x\in X$;

  2. cmrmn(ii)

$S_{n}^{A}(x)\to x$ in $(X,\|\cdot\|)$ for all $x\in X$;

  3. cmrmn(iii)

$(S_{n}^{A})^{*}(\phi)\to\phi$ in $(X^{*},w^{*})$ for all $\phi\in X^{*}$;

  4. cmrmn(iv)

$(S_{n}^{A})^{*}(\phi)\to\phi$ in $(X^{*},w^{*})$ for all $\phi\in Z$;

  5. cmrmn(v)

$(S_{n}^{A})^{*}(\phi)\to\phi$ in $(X^{*},\|\cdot\|)$ for all $\phi\in Z$.

If, further $X$, is reflexive, then these are equivalent to:

  1. cmrmn(vi)

$(S_{n}^{A})^{*}(\phi)\to\phi$ in $(X^{*},\|\cdot\|)$ for all $\phi\in X^{*}$.

###### Proof.

By (3.1), for each $n\geq 0$, there exists $K_{n}\geq 0$ such that

$\sum_{k>K_{n}}|a_{nk}|\|P_{k}\|<2^{-n}.$

Define $T_{n}^{A}:X\to X$ by

$T_{n}^{A}:=\sum_{k=0}^{K_{n}}a_{nk}P_{k}.$

Clearly we have $\|S_{n}^{A}-T_{n}^{A}\|<2^{-n}$ for all $n$, so each of the
statements (i)–(vi) holds iff it holds with $S_{n}^{A}$ replaced by
$T_{n}^{A}$.

The operators $T_{n}^{A}$ are bounded, finite-rank operators on $X$. The
images of $T_{n}^{A}$ and $(T_{n}^{A})^{*}$ are given by

(3.2) $\displaystyle T_{n}^{A}(X)$
$\displaystyle=\operatorname{span}\\{e_{k}:1\leq k\leq K_{n},~{}a_{nk}\neq
0\\},$ (3.3) $\displaystyle(T_{n}^{A})^{*}(X^{*})$
$\displaystyle=\operatorname{span}\\{\psi_{k}:1\leq k\leq K_{n},~{}a_{nk}\neq
0\\}.$

It follows that $T_{n}^{A}T_{m}^{A}(X)\subset T_{m}^{A}(X)$ and
$(T_{n}^{A})^{*}(T_{m}^{A})^{*}(X^{*})\subset(T_{m}^{A})^{*}(X^{*})$ for all
$m,n$. Thus Theorem 2.1 applies.

We claim that, for each $k$, there is at least one $n$ such that $a_{nk}\neq
0$. Indeed, if $a_{nk}=0$ for all $n$, then $T_{n}^{A}(e_{k})=0$ and
$(T_{n}^{A})^{*}(\psi_{k})=0$ for all $n$, and so (under the relevant
assumption (i)–(vi)), either $e_{k}=0$ or $\psi_{k}=0$. Neither of these can
be true, since we are assuming that $\langle e_{k},\psi_{k}\rangle\neq 0$.

It follows that, in the notation of Theorem 2.1,

$\displaystyle Y$ $\displaystyle:=\overline{\operatorname{span}(\cup_{m\geq
0}(T_{m}^{A})(X))}=\overline{\operatorname{span}\\{e_{k}:k\geq 0\\}},$
$\displaystyle Z$ $\displaystyle:=\overline{\operatorname{span}(\cup_{m\geq
0}(T_{m}^{A})^{*}(X^{*}))}=\overline{\operatorname{span}\\{\psi_{k}:k\geq
0\\}}.$

In particular, this reconciles the definition of $Z$ given in the statement of
Theorem 3.1 with that in Theorem 2.1.

From Theorem 2.1, applied with $A=A^{0}$, we deduce that $Y=X$ and that $Z$ is
$w^{*}$-sequentially dense in $X^{*}$. Reapplying Theorem 2.1 with a general
$A$ now gives the result. ∎

######  Remarks.

(1) The implications (i)$\iff$(ii)$\iff$(iii)$\Longrightarrow$(iv)$\iff$(v)
hold without the assumption about $A^{0}$. The existence of $A^{0}$ is needed
only for the implication (iv)$\Longrightarrow$(iii).

(2) In the concrete examples that we shall treat below, the existence of
$A^{0}$ will be an obvious consequence of some version of Fejér’s theorem.

(3) However, there do exist examples where there exists no matrix $A^{0}$
satisfying (ii). For instance, this is the case whenever the Banach space $X$
fails to have the bounded approximation property (see [13]). The article [10]
contains another such example, in which $X$ is actually a Hilbert space.

###  3.3. Limitation theorems

Let $A=(a_{nk})_{n,k\geq 0}$ be an infinite scalar matrix satisfying (3.1). We
say that $A$ admits a left inverse if there exists a scalar matrix
$B=(b_{jn})_{j,n\geq 0}$ such that

(3.4) $\sum_{n\geq 0}|b_{jn}|<\infty\quad(j\geq
0)\qquad\text{and}\qquad\sum_{n\geq 0}b_{jn}a_{nk}=\begin{cases}1,&j=k,\\\
0,&j\neq k.\end{cases}$

######  Theorem 3.2.

Suppose that $S_{n}^{A}(x)\to x$ (weakly or in norm) for all $x\in X$, and
that $A$ admits a left inverse $B$. Then, writing $B_{j}:=\sum_{n\geq
0}|b_{jn}|$, we have

$\frac{\|e_{j}\|\|\psi_{j}\|}{|\langle
e_{j},\psi_{j}\rangle|}=O(B_{j})\quad(j\to\infty).$

######  Remark.

This theorem is of interest when $\|e_{j}\|\|\psi_{j}\|/|\langle
e_{j},\psi_{j}\rangle|$ grows with $j$, since it then places limitations on
possible matrices $A$ for which summability holds. Hardy [8, p.57] calls such
results limitation theorems.

###### Proof.

For each $j\geq 0$, we have

$\displaystyle\sum_{n\geq 0}b_{jn}S_{n}^{A}$ $\displaystyle=\sum_{n\geq
0}b_{jn}\Bigl{(}\sum_{k\geq 0}a_{nk}P_{k}\Bigr{)}$ $\displaystyle=\sum_{k\geq
0}\Bigl{(}\sum_{n\geq 0}b_{jn}a_{nk}\Bigr{)}P_{k}=\sum_{k\geq
0}\delta_{jk}P_{k}=P_{j},$

the exchange of sums being justified by absolute convergence in (3.1) and
(3.4). Now as $S_{n}^{A}(x)\to x$ for all $x\in X$, we have
$\sup_{n}\|S_{n}^{A}(x)\|<\infty$ for each $x\in X$, and hence, by the
Banach–Steinhaus theorem, $\sup_{n}\|S_{n}^{A}\|<\infty$. It follows that

$\|P_{j}\|=\Bigl{\|}\sum_{n\geq 0}b_{jn}S_{n}^{A}\Bigr{\|}\leq\sum_{n\geq
0}|b_{jn}|\|S_{n}^{A}\|\leq B_{j}\sup_{n}\|S_{n}^{A}\|,$

in other words, $\|P_{j}\|=O(B_{j})$. Finally, since
$\|P_{j}\|=\|e_{j}\|\|\psi_{j}\|/|\langle e_{j},\psi_{j}\rangle|$, the result
follows. ∎

In practice, the matrix $B$ is not known explicitly, so estimating $B_{j}$ may
be problematic. Here is one case where we can do it.

We denote by $W^{+}({\mathbb{D}})$ the holomorphic Wiener algebra, namely

$W^{+}({\mathbb{D}}):=\Bigl{\\{}f(z)=\sum_{k\geq
0}\widehat{f}(k)z^{k}:\sum_{k\geq 0}|\widehat{f}(k)|<\infty\Bigr{\\}}.$

######  Theorem 3.3.

Let $f\in W^{+}({\mathbb{D}})$ with $f(0)\neq 0$, and let $(\gamma_{n})_{n\geq
0}$ be increasing sequence in $(0,\infty)$. Let $A=(a_{nk})$ be the lower-
triangular matrix with entries given by

$a_{nk}:=\gamma_{n}^{-1}\widehat{(1/f)}(n-k)\quad(0\leq k\leq n).$

Then $A$ has a left inverse $B$, where $B=(b_{jn})$ is a lower-triangular
matrix and

$\sum_{n\geq 0}|b_{jn}|\asymp\gamma_{j}\quad(j\to\infty).$

###### Proof.

Let $B$ be the lower-triangular matrix with entries $(b_{jn})$ given by

$b_{jn}:=\gamma_{n}\widehat{f}(j-n)\quad(0\leq n\leq j).$

For each fixed $j$, there are only finitely many $n$ for which $b_{jn}\neq 0$,
so the first condition in (3.4) is clearly satisfied. As for the second
condition, we have

$\displaystyle\sum_{n\geq 0}b_{jn}a_{nk}$ $\displaystyle=\sum_{n\geq
0}\gamma_{n}\widehat{f}(j-n)\gamma_{n}^{-1}\widehat{(1/f)}(n-k)$
$\displaystyle=\sum_{\begin{subarray}{c}p,q\geq 0\\\
p+q=j-k\end{subarray}}\widehat{f}(p)\widehat{(1/f)}(q)=\widehat{1}(j-k)=\delta_{jk}.$

Thus $B$ is indeed a left inverse of $A$. Further, we have

$\sum_{n\geq
0}|b_{jn}|=\sum_{n=0}^{j}\gamma_{n}|\widehat{f}(j-n)|\leq\gamma_{j}\sum_{\ell\geq
0}|\widehat{f}(\ell)|$

and

$\sum_{n\geq
0}|b_{jn}|=\sum_{n=0}^{j}\gamma_{n}|\widehat{f}(j-n)|\geq\gamma_{j}|\widehat{f}(0)|,$

By assumption, we have $\sum_{\ell\geq 0}|\widehat{f}(\ell)|<\infty$ and
$|\widehat{f}(0)|>0$. It follows that $\sum_{n\geq
0}|b_{jn}|\asymp\gamma_{j}$, as claimed. ∎

######  Remark.

In this case, it is easy to see that in fact $B$ is a two-sided inverse of
$A$.

We illustrate these results by applying them to one particular family of
summability methods, namely the Cesàro means. Given $\alpha>-1$, let
$A=(a_{nk})_{n,k\geq 0}$ be the lower-triangular matrix defined by

(3.5) $a_{nk}:=\binom{n}{k}\Big{/}\binom{n+\alpha}{k}\quad(0\leq k\leq n).$

(As usual, binomial coefficients with non-integer arguments are defined using
the Gamma function.) With this choice of $A$, we write

$\sigma_{n}^{\alpha}(x):=S_{n}^{A}(x)\quad(x\in X).$

In particular, we write $s_{n}$ for $\sigma_{n}^{0}$ and $\sigma_{n}$ for
$\sigma_{n}^{1}$.

We remark that, if $-1<\alpha\leq\beta$, then $\sigma_{n}^{\beta}$ includes
$\sigma_{n}^{\alpha}$ in the sense that, if $\sigma_{n}^{\alpha}(x)\to x$ for
some $x\in X$, then also $\sigma_{n}^{\beta}(x)\to x$. (For scalars this is
well known; for Banach spaces, it follows by [11, Theorem 5.1].)

The following limitation theorem is an abstract version of [8, Theorem 46].

######  Theorem 3.4.

Let $\alpha\geq 0$. If $\sigma_{n}^{\alpha}(x)\to x$ (weakly or in norm) for
all $x\in X$, then

$\frac{\|e_{j}\|\|\psi_{j}\|}{|\langle
e_{j},\psi_{j}\rangle|}=O(j^{\alpha})\quad(j\to\infty).$

###### Proof.

Let $A$ be the Cesàro matrix defined by (3.5). A computation gives

$a_{nk}=\binom{n}{k}\Big{/}\binom{n+\alpha}{k}=\binom{n-k+\alpha}{\alpha}\Bigl{/}\binom{n+\alpha}{\alpha}=\gamma_{n}^{-1}\widehat{g}(n-k),$

where $(\gamma_{n})$ is the increasing sequence given by

$\gamma_{n}:=\binom{n+\alpha}{\alpha},$

and where $\widehat{g}(m)$ are the Taylor coefficients of the function

$g(z):=\sum_{m=0}^{\infty}\binom{m+\alpha}{\alpha}z^{m}=(1-z)^{-\alpha-1}.$

Clearly $g=1/f$, where $f(z):=(1-z)^{\alpha+1}$. Since the Taylor coefficients
of $f$ satisfy

$\sum_{k\geq 1}|\widehat{f}(k)|=\sum_{k\geq
1}\Bigl{|}\frac{\widehat{(f^{\prime})}(k-1)}{k}\Bigr{|}\leq\Bigl{(}\sum_{k\geq
1}\frac{1}{k^{2}}\Bigr{)}^{1/2}\|f^{\prime}\|_{H^{2}}<\infty,$

we have $f\in W^{+}({\mathbb{D}})$, and Theorem 3.3 applies. We deduce that
$A$ has a lower-triangular inverse $B=(b_{jn})$ such that

$\sum_{n\geq
0}|b_{jn}|=O(\gamma_{j})=O\Bigl{(}\binom{j+\alpha}{\alpha}\Bigr{)}=O(j^{\alpha})\quad(j\to\infty),$

the last equality by Stirling’s formula. The result now follows by applying
Theorem 3.2. ∎

##  4\. Applications in spaces of continuous functions

###  4.1. Fourier series

Probably the best-known applications of summability are to Fourier series, so,
for our first example, we see what the abstract theory developed in the
previous two sections tells us about this case.

Let ${\mathbb{T}}$ denote the unit circle. We write $C({\mathbb{T}})$ for the
space of complex-valued continuous functions on ${\mathbb{T}}$, with the usual
sup-norm $\|f\|_{\infty}:=\sup_{\mathbb{T}}|f|$. The dual space of
$C({\mathbb{T}})$ may be identified with $M({\mathbb{T}})$, the space of
finite complex Borel measures on ${\mathbb{T}}$, the duality being given by

$\langle
f,\mu\rangle:=\int_{\mathbb{T}}f(\zeta)\,d\mu(\overline{\zeta})\quad(f\in
C({\mathbb{T}}),~{}\mu\in M({\mathbb{T}})).$

Under this pairing, $M({\mathbb{T}})$ inherits the norm of
$C({\mathbb{T}})^{*}$, which is just the total-variation norm.

The absolutely continuous measures on ${\mathbb{T}}$ form a closed subspace of
$M({\mathbb{T}})$, which can be identified with $L^{1}({\mathbb{T}})$ via
$g\leftrightarrow g(e^{it})dt/2\pi$. The restriction of the total variation
norm to $L^{1}({\mathbb{T}})$ is just the usual $L^{1}$ norm.

We now proceed to apply the theory developed in §3. Let $X:=C({\mathbb{T}})$,
and for $k\in{\mathbb{Z}}$ let $e_{k}:=e^{ikt}$ and
$\psi_{k}:=e^{ikt}\,dt/2\pi$. (Here the index $k$ runs through all the
integers rather than just the positive integers, but this creates no
problems.) Then $\|e_{k}\|_{\infty}=\|\psi_{k}\|_{1}=1$ and $\langle
e_{k},\psi_{k}\rangle=1$ for all $k\in{\mathbb{Z}}$.

Let $A=(a_{nk})_{n\geq 0,k\in{\mathbb{Z}}}$ be an infinite matrix of complex
scalars such that $\sum_{k\in{\mathbb{Z}}}|a_{nk}|<\infty$ for each $n\geq 0$.
Then, for each $n\geq 0$, we have

$\displaystyle S_{n}^{A}(f)$
$\displaystyle=\sum_{k\in{\mathbb{Z}}}a_{nk}\frac{\langle
f,\psi_{k}\rangle}{\langle
e_{k},\psi_{k}\rangle}e_{k}=\sum_{k\in{\mathbb{Z}}}a_{nk}\widehat{f}(k)e^{ikt}\quad(f\in
C({\mathbb{T}})),$ $\displaystyle(S_{n}^{A})^{*}(\mu)$
$\displaystyle=\sum_{k\in{\mathbb{Z}}}a_{nk}\frac{\langle
e_{k},\mu\rangle}{\langle
e_{k},\psi_{k}\rangle}\psi_{k}=\sum_{k\in{\mathbb{Z}}}a_{nk}\widehat{\mu}(k)e^{ikt}\,\frac{dt}{2\pi}\quad(\mu\in
M({\mathbb{T}})).$

By Fejér’s theorem, we have $\|\sigma_{n}f-f\|_{\infty}\to 0$ as $n\to\infty$
for all $f\in C({\mathbb{T}})$, so there exists at least one matrix $A^{0}$
for which $\|S_{n}^{A^{0}}(f)-f\|_{\infty}\to 0$. Thus Theorem 3.1 applies. In
the notation of Theorem 3.1, $Z$ is the norm-closure in $M({\mathbb{T}})$ of
$\operatorname{span}\\{e^{ikt}dt/2\pi:k\in{\mathbb{Z}}\\}$, which is exactly
$L^{1}({\mathbb{T}})$. Thus we obtain the following theorem.

######  Theorem 4.1.

Let $(a_{nk})_{n\geq 0,k\in{\mathbb{Z}}}$ be an infinite matrix of complex
scalars such that $\sum_{k\in{\mathbb{Z}}}|a_{nk}|<\infty$ for each $n\geq 0$.
Then the following statements are equivalent:

  1. cmrmn(i)

$\sum_{k\in{\mathbb{Z}}}a_{nk}\widehat{f}(k)e^{ikt}\to f$ in
$(C({\mathbb{T}}),w)$, for all $f\in C({\mathbb{T}})$;

  2. cmrmn(ii)

$\sum_{k\in{\mathbb{Z}}}a_{nk}\widehat{f}(k)e^{ikt}\to f$ in
$(C({\mathbb{T}}),\|\cdot\|_{\infty})$), for all $f\in C({\mathbb{T}})$;

  3. cmrmn(iii)

$\sum_{k\in{\mathbb{Z}}}a_{nk}\widehat{\mu}(k)e^{ikt}dt/2\pi\to\mu$ in
$(M({\mathbb{T}}),w^{*})$, for all $\mu\in M({\mathbb{T}})$;

  4. cmrmn(iv)

$\sum_{k\in{\mathbb{Z}}}a_{nk}\widehat{g}(k)e^{ikt}dt/2\pi\to
g(e^{it})dt/2\pi$ in $(M({\mathbb{T}}),w^{*})$ for all $g\in
L^{1}({\mathbb{T}})$;

  5. cmrmn(v)

$\sum_{k\in{\mathbb{Z}}}a_{nk}\widehat{g}(k)e^{ikt}\to g$ in
$(L^{1}({\mathbb{T}}),\|\cdot\|_{1})$, for all $g\in L^{1}({\mathbb{T}})$.

We now use this theorem to deduce some classical results about Cesàro
summation of Fourier series. Since the summation index $k$ runs over
${\mathbb{Z}}$ rather than ${\mathbb{Z}}^{+}$, the definition of
$\sigma_{n}^{\alpha}$ needs to be modified accordingly, taking
$a_{nk}:=\binom{n}{|k|}/\binom{n+\alpha}{|k|}$ for $|k|\leq n$ and $a_{nk}:=0$
for $|k|>n$. As usual, we write $s_{n}$ for $\sigma_{n}^{0}$ and $\sigma_{n}$
for $\sigma_{n}^{1}$ (this was already implicit when we quoted Fejér’s theorem
above).

######  Theorem 4.2.

  1. cmrmn(1)

If $\mu=\delta_{1}$, then $s_{n}(\mu)\not\to\mu$ in $(M({\mathbb{T}}),w^{*})$.

  2. cmrmn(2)

There exists $f\in C({\mathbb{T}})$ such that $s_{n}(f)\not\to f$ in
$(C({\mathbb{T}}),w)$.

  3. cmrmn(3)

There exists $g\in L^{1}({\mathbb{T}})$ such that $s_{n}(g)\not\to g$ in
$(M({\mathbb{T}}),w^{*})$.

###### Proof.

(1) If $\mu=\delta_{1}$, then $\widehat{\mu}(k)=1$ for all $k\in{\mathbb{Z}}$,
so

$s_{n}(\mu)=\sum_{k=-n}^{n}e^{ikt}\frac{dt}{2\pi}=D_{n}(t)\,\frac{dt}{2\pi},$

where $D_{n}(t)$ is the Dirichlet kernel. We know that

$\|s_{n}(\mu)\|_{M({\mathbb{T}})}=\|D_{n}\|_{1}\asymp\log n\to\infty,$

so $(s_{n}(\mu))$ is not weak*-convergent.

Parts (2) and (3) now follow from (1) by applying Theorem 4.1. ∎

######  Remark.

Part (2) is a weak form of a famous result of du Bois-Reymond, who showed that
the Fourier series of a continuous function may diverge at a point.

######  Theorem 4.3.

Let $\alpha>0$.

  1. cmrmn(1)

$\|\sigma_{n}^{\alpha}(f)-f\|_{\infty}\to 0$ for all $f\in C({\mathbb{T}})$.

  2. cmrmn(2)

$\|\sigma_{n}^{\alpha}(g)-g\|_{1}\to 0$ for all $g\in L^{1}({\mathbb{T}})$.

  3. cmrmn(3)

$\sigma_{n}^{\alpha}(\mu)\to\mu$ in $(M({\mathbb{T}}),w^{*})$ for all $\mu\in
M({\mathbb{T}})$.

###### Proof.

Part (1) is a classical result of M. Riesz [14]. Parts (2) and (3) follow,
using Theorem 4.1. ∎

###  4.2. The disk algebra

The disk algebra $A({\mathbb{D}})$ consists of those holomorphic functions on
the open unit disk ${\mathbb{D}}$ that have a continuous extension to
$\overline{{\mathbb{D}}}$. It is a Banach space (indeed a Banach algebra) with
respect to the sup-norm.

By the maximum principle, the map $f\mapsto f|_{\mathbb{T}}$ is an isometry of
$A({\mathbb{D}})$ into $C({\mathbb{T}})$, so $A({\mathbb{D}})$ can be
identified with a closed subspace $A$ of $C({\mathbb{T}})$. In fact $A=\\{f\in
C({\mathbb{T}}):\widehat{f}(k)=0~{}\forall k<0\\}$. Therefore the dual of
$A({\mathbb{D}})$ may be identified with the quotient
$M({\mathbb{T}})/A^{\perp}$, where

$A^{\perp}:=\Bigl{\\{}\mu\in
M({\mathbb{T}}):\int_{\mathbb{T}}f\,d\mu=0~{}\forall f\in A\Bigr{\\}}.$

By the F. & M. Riesz theorem, if $\mu\in A^{\perp}$, then $\mu$ is absolutely
continuous with respect to Lebesgue measure on ${\mathbb{T}}$, say
$\mu=h\,dt/2\pi$. The condition that $h\,dt/2\pi\in A^{\perp}$ is equivalent
to $h\in\overline{H^{1}_{0}}$, where

$\overline{H}_{0}^{1}:=\\{h\in L^{1}({\mathbb{T}}):\widehat{h}(k)=0~{}\forall
k\geq 0\\}.$

Thus we can identify the dual space $A({\mathbb{D}})^{*}$ with
$M({\mathbb{T}})/\overline{H^{1}_{0}}$.

There is another way to express this duality, using Cauchy transforms. Given
$\mu\in M({\mathbb{T}})$, we define its Cauchy transform
$K\mu:{\mathbb{D}}\to{\mathbb{C}}$ by

$K\mu(z):=\int_{\mathbb{T}}\frac{d\mu(\zeta)}{1-\overline{\zeta}z}\quad(z\in{\mathbb{D}}).$

Notice that $K\mu\equiv 0\iff\mu\in\overline{H_{0}^{1}}$. Hence the map
$[\mu]\mapsto K\mu$ is a linear isomorphism of
$M({\mathbb{T}})/\overline{H_{0}^{1}}$ onto ${\mathcal{K}}$, where

${\mathcal{K}}:=\\{K\mu:\mu\in M({\mathbb{T}})\\}.$

We endow ${\mathcal{K}}$ with the norm that makes this isomorphism an
isometry, namely

$\|K\mu\|_{\mathcal{K}}:=\|[\mu]\|_{M({\mathbb{T}})/\overline{H_{0}^{1}}}=\operatorname{dist}(\mu,\overline{H_{0}^{1}})\quad(\mu\in
M({\mathbb{T}})).$

Thus, finally, the dual of $A({\mathbb{D}})$ may be identified with
${\mathcal{K}}$, the duality pairing being given by

$\langle
f,K\mu\rangle:=\int_{\mathbb{T}}f(\zeta)\,d\mu(\overline{\zeta})=\lim_{r\to
1^{-}}\sum_{k=0}^{\infty}\widehat{f}(k)\widehat{\mu}(k)r^{k}\quad(f\in
A({\mathbb{D}}),~{}K\mu\in{\mathcal{K}}).$

We now apply the theory developed in §3. Let $X:=A({\mathbb{D}})$ and
$X^{*}={\mathcal{K}}$. For $k\geq 0$, we define $e_{k}\in A({\mathbb{D}})$ by
$e_{k}(z):=z^{k}$, and $\psi_{k}\in{\mathcal{K}}$ by

$\psi_{k}(z):=K\Bigl{(}e^{ikt}\,\frac{dt}{2\pi}\Bigr{)}(z)=z^{k}.$

It is easily checked that $\|e_{k}\|_{\infty}=\|\psi_{k}\|_{\mathcal{K}}=1$
and that $\langle e_{k},\psi_{k}\rangle=1$ for all $k$.

Let $A=(a_{nk})_{n,k\geq 0}$ be an infinite matrix of complex scalars such
that $\sum_{k\geq 0}|a_{nk}|<\infty$ for each $n\geq 0$. Then, for each $n\geq
0$, we have

$\displaystyle S_{n}^{A}(f)$ $\displaystyle=\sum_{k\geq 0}a_{nk}\frac{\langle
f,\psi_{k}\rangle}{\langle e_{k},\psi_{k}\rangle}e_{k}=\sum_{k\geq
0}a_{nk}\widehat{f}(k)z^{k}\quad(f\in A({\mathbb{D}})),$
$\displaystyle(S_{n}^{A})^{*}(K\mu)$ $\displaystyle=\sum_{k\geq
0}a_{nk}\frac{\langle e_{k},K\mu\rangle}{\langle
e_{k},\psi_{k}\rangle}\psi_{k}=\sum_{k\geq
0}a_{nk}\widehat{\mu}(k)z^{k}\quad(K\mu\in{\mathcal{K}}).$

Just as in the case of $C({\mathbb{T}})$, Fejér’s theorem implies that
$\|\sigma_{n}f-f\|_{\infty}\to 0$ as $n\to\infty$ for all $f\in
A({\mathbb{D}})$, so there exists at least one matrix $A^{0}$ for which
$\|S_{n}^{A^{0}}(f)-f\|_{\infty}\to 0$. Thus Theorem 3.1 applies. In the
notation of Theorem 3.1, $Z$ is the norm-closure in ${\mathcal{K}}$ of
$\operatorname{span}\\{K(e^{ikt}dt/2\pi):k\geq 0\\}$, which is
$L^{1}({\mathbb{T}})/\overline{H^{1}_{0}}$. Thus we obtain the following
theorem.

######  Theorem 4.4.

Let $(a_{nk})_{n,k\geq 0}$ be an infinite matrix of complex scalars such that
$\sum_{k\in{\mathbb{Z}}}|a_{nk}|<\infty$ for each $n\geq 0$. Then the
following statements are equivalent:

  1. cmrmn(i)

$\sum_{k\geq 0}a_{nk}\widehat{f}(k)z^{k}\to f$ in $(A({\mathbb{D}}),w)$, for
all $f\in A({\mathbb{D}})$;

  2. cmrmn(ii)

$\sum_{k\geq 0}a_{nk}\widehat{f}(k)z^{k}\to f$ in
$(A({\mathbb{D}}),\|\cdot\|_{\infty})$ for all $f\in A({\mathbb{D}})$;

  3. cmrmn(iii)

$\sum_{k\geq 0}a_{nk}\widehat{\mu}(k)z^{k}\to K\mu$ in
$({\mathcal{K}},w^{*})$, for all $\mu\in M({\mathbb{T}})$;

  4. cmrmn(iv)

$\sum_{k\geq 0}a_{nk}\widehat{g}(k)z^{k}\to Kg$ in $({\mathcal{K}},w^{*})$,
for all $g\in L^{1}({\mathbb{T}})$;

  5. cmrmn(v)

$\sum_{k\geq 0}a_{nk}\widehat{g}(k)z^{k}\to g$ in
$(L^{1}/\overline{H^{1}_{0}},\|\cdot\|_{L^{1}/\overline{H^{1}_{0}}})$, for all
$g\in L^{1}({\mathbb{T}})$.

##  5\. Application to Hardy spaces and Bergman spaces

###  5.1. Hardy spaces, $\operatorname{BMOA}$ and $\operatorname{VMOA}$

We begin by reviewing the definitions and some basic facts about these spaces.
All the details can be found in [20, Chapter 9].

For $1\leq p<\infty$, the Hardy space $H^{p}$ is defined as the set of
$f\in\operatorname{Hol}({\mathbb{D}})$ such that

$\|f\|_{H^{p}}:=\sup_{r<1}\Bigl{(}\frac{1}{2\pi}\int_{0}^{2\pi}|f(re^{i\theta})|^{p}\,d\theta\Bigr{)}^{1/p}<\infty.$

Also $H^{\infty}$ is the set of bounded holomorphic functions on
${\mathbb{D}}$, with

$\|f\|_{H^{\infty}}:=\sup_{\mathbb{D}}|f|.$

The space $\operatorname{BMOA}$ of holomorphic functions of bounded mean
oscillation can be characterized as the space of $f\in H^{2}$ such that

$\|f\|_{\operatorname{BMOA}}:=|f(0)|+\sup_{{\mathbb{D}}}(P|f|^{2}-|f|^{2})^{1/2}<\infty,$

where $P|f|^{2}$ denotes the Poisson integral of $|f|^{2}$. The space
$\operatorname{VMOA}$ of holomorphic functions of vanishing mean oscillation
is the closed subspace of $\operatorname{BMOA}$ consisting of those
$f\in\operatorname{BMOA}$ such that $(P|f|^{2}-|f|^{2})(z)\to 0$ as $|z|\to
1$.

All the spaces $H^{p},\operatorname{VMOA},\operatorname{BMOA}$ are Banach
spaces and contain the polynomials. Polynomials are dense in $H^{p}~{}(1\leq
p<\infty)$ and in $\operatorname{VMOA}$, but not in $H^{\infty}$ or
$\operatorname{BMOA}$, since neither of the latter is separable.

We have the following identifications of dual spaces (up to isomorphism):
$(\operatorname{VMOA})^{*}\cong H^{1}$ and
$(H^{1})^{*}\cong\operatorname{BMOA}$. The pairings are given by

$\displaystyle\langle f,g\rangle$ $\displaystyle:=\lim_{r\to
1^{-}}\frac{1}{2\pi}\int_{0}^{2\pi}f(re^{i\theta})g(re^{-i\theta})\,d\theta\quad(f\in\operatorname{VMOA},~{}g\in
H^{1}),$ $\displaystyle\langle g,h\rangle$ $\displaystyle:=\lim_{r\to
1^{-}}\frac{1}{2\pi}\int_{0}^{2\pi}g(re^{i\theta})h(re^{-i\theta})\,d\theta\quad(g\in
H^{1},~{}h\in\operatorname{BMOA}).$

We now proceed to apply the theory developed in §3. Let $X:=H^{1}$, let
$e_{k}:=z^{k}\in H^{1}$ and let $\psi_{k}:=z^{k}\in\operatorname{BMOA}$. Then
$\|e_{k}\|=1$, $\|\psi_{k}\|\asymp 1$ and $\langle e_{k},\psi_{k}\rangle=1$
for all $k\geq 0$.

Let $(a_{nk})_{n,k\geq 0}$ be a matrix of complex scalars such that
$\sum_{k}|a_{nk}|<\infty$ for each $n$. Then

$\displaystyle S_{n}^{A}(g)$ $\displaystyle=\sum_{k\geq 0}a_{nk}\frac{\langle
g,\psi_{k}\rangle}{\langle e_{k},\psi_{k}\rangle}e_{k}=\sum_{k\geq
0}a_{nk}\widehat{g}(k)z^{k}\quad(g\in H^{1}),$
$\displaystyle(S_{n}^{A})^{*}(h)$ $\displaystyle=\sum_{k\geq
0}a_{nk}\frac{\langle e_{k},h\rangle}{\langle
e_{k},\psi_{k}\rangle}\psi_{k}=\sum_{k\geq
0}a_{nk}\widehat{h}(k)z^{k}\quad(h\in\operatorname{BMOA}).$

By an appropriate version of Fejér’s theorem, $\|\sigma_{n}(g)-g\|_{H^{1}}\to
0$ for all $g\in H^{1}$. Theorem 3.1 therefore applies. In the notation of
Theorem 3.1, $Z$ is the norm-closure in $\operatorname{BMOA}$ of
$\operatorname{span}\\{z^{k}:k\geq 0\\}$, which is exactly
$\operatorname{VMOA}$. We thus obtain the following result.

######  Theorem 5.1.

Let $(a_{nk})_{n,k\geq 0}$ be an infinite matrix of complex scalars such that
$\sum_{k}|a_{nk}|<\infty$ for each $n$. Then the following statements are
equivalent:

  1. cmrmn(i)

$\sum_{k\geq 0}a_{nk}\widehat{g}(k)z^{k}\to g$ in $(H^{1},w)$, for all $g\in
H^{1}$;

  2. cmrmn(ii)

$\sum_{k\geq 0}a_{nk}\widehat{g}(k)z^{k}\to g$ in $(H^{1},\|\cdot\|_{H^{1}})$,
for all $g\in H^{1}$;

  3. cmrmn(iii)

$\sum_{k\geq 0}a_{nk}\widehat{h}(k)z^{k}\to h$ in
$(\operatorname{BMOA},w^{*})$, for all $h\in\operatorname{BMOA}$;

  4. cmrmn(iv)

$\sum_{k\geq 0}a_{nk}\widehat{f}(k)z^{k}\to f$ in $(\operatorname{VMOA},w)$,
for all $f\in\operatorname{VMOA}$;

  5. cmrmn(v)

$\sum_{k\geq 0}a_{nk}\widehat{f}(k)z^{k}\to f$ in
$(\operatorname{VMOA},\|\cdot\|_{\operatorname{BMOA}})$, for all
$f\in\operatorname{VMOA}$.

######  Remark.

Explicitly, statement (iii) means that, for all $h\in\operatorname{BMOA}$ and
all $g\in H^{1}$,

$\lim_{n\to\infty}\lim_{r\to 1^{-}}\sum_{k\geq
0}a_{nk}r^{k}\widehat{g}(k)\widehat{h}(k)=\lim_{r\to
1^{-}}\frac{1}{2\pi}\int_{0}^{2\pi}g(re^{i\theta})h(re^{-i\theta})\,d\theta.$

We now specialize to the case of Cesàro means.

######  Theorem 5.2.

  1. cmrmn(1)

There exists $g\in H^{1}$ such that $s_{n}(g)\not\to g$ in $(H^{1},w)$.

  2. cmrmn(2)

There exists $f\in\operatorname{VMOA}$ such that $s_{n}(f)\not\to f$ in
$(\operatorname{VMOA},w)$.

###### Proof.

It is well known that there exists $g\in H^{1}$ such that
$\|s_{n}(g)-g\|_{H1}\not\to 0$. This follows easily from the fact that the
Riesz projection $P_{+}:L^{1}({\mathbb{T}})\to H^{1}$ is unbounded (see e.g.
[6, Ch. III, §1]). Parts (1) and (2) both follow by applying the equivalences
in Theorem 5.1. ∎

######  Remark.

Using much the same idea, Zhu has previously shown that there exists
$f\in\operatorname{VMOA}$ such that
$\|s_{n}(f)-f\|_{\operatorname{BMOA}}\not\to 0$ (see [19, Corollary 5]).

######  Theorem 5.3.

Let $\alpha>0$.

  1. cmrmn(1)

$\|\sigma_{n}^{\alpha}(g)-g\|_{H^{1}}\to 0$ for all $g\in H^{1}$.

  2. cmrmn(2)

$\sigma_{n}^{\alpha}(h)\to h$ in $(\operatorname{BMOA},w^{*})$ for all
$h\in\operatorname{BMOA}$.

  3. cmrmn(3)

$\|\sigma_{n}^{\alpha}(f)-f\|_{\operatorname{BMOA}}\to 0$ for all
$f\in\operatorname{VMOA}$.

###### Proof.

Part (1) is a classical result of Hardy [7]. Parts (2) and (3) follow, using
Theorem 5.1. ∎

###  5.2. Bergman and Bloch spaces

Our development parallels that in §5.1. Once again, we begin by reviewing the
definitions and some basic facts about these spaces. The details are in [20,
Chapters 4 and 5].

For $1\leq p<\infty$, the Bergman space $A^{p}$ is defined as the space of
$f\in\operatorname{Hol}({\mathbb{D}})$ such that

$\|f\|_{A^{p}}:=\Bigl{(}\frac{1}{\pi}\int_{\mathbb{D}}|f(z)|^{p}\,dA(z)\Bigr{)}^{1/p}<\infty,$

where $dA$ denotes area measure on ${\mathbb{D}}$.

The Bloch space ${\mathcal{B}}$ consists of those
$f\in\operatorname{Hol}({\mathbb{D}})$ such that

$\|f\|_{{\mathcal{B}}}:=|f(0)|+\sup_{{\mathbb{D}}}(1-|z|^{2})|f^{\prime}(z)|<\infty.$

The little Bloch space ${\mathcal{B}}_{0}$ is the closed subspace of
${\mathcal{B}}$ consisting of those functions $f\in{\mathcal{B}}$ such that
$(1-|z|^{2})|f^{\prime}(z)|\to 0$ as $|z|\to 1$.

All the spaces $A^{p},{\mathcal{B}},{\mathcal{B}}_{0}$ are Banach spaces and
contain the polynomials. Polynomials are dense in $A^{p}~{}(1\leq p<\infty)$
and in ${\mathcal{B}}_{0}$, but not in ${\mathcal{B}}$, since the latter is
not separable.

We have the following identifications of dual spaces (up to isomorphism):
$({\mathcal{B}}_{0})^{*}\cong A^{1}$ and $(A^{1})^{*}\cong{\mathcal{B}}$. The
pairings are given by

$\displaystyle\langle f,g\rangle$ $\displaystyle:=\lim_{r\to
1^{-}}\frac{1}{\pi}\int_{|z|<r}f(z)g(\overline{z})\,dA(z)\quad(f\in{\mathcal{B}}_{0},~{}g\in
A^{1}),$ $\displaystyle\langle g,h\rangle$ $\displaystyle:=\lim_{r\to
1^{-}}\frac{1}{\pi}\int_{|z|<r}g(z)h(\overline{z})\,dA(z)\quad(g\in
A^{1},~{}h\in{\mathcal{B}}).$

Once again, we apply the theory developed in §3. Let $X:=A^{1}$, let
$e_{k}:=z^{k}\in A^{1}$ and let $\psi_{k}:=z^{k}\in{\mathcal{B}}$. Then
$\|e_{k}\|_{A^{1}}\asymp 1/(k+1)$ and $\|\psi_{k}\|_{\mathcal{B}}\asymp 1$,
and $|\langle e_{k},\psi_{k}\rangle|\asymp 1/(k+1)$, where the implied
constants are independent of $k$.

Let $(a_{nk})_{n,k\geq 0}$ be a matrix of complex scalars such that
$\sum_{k}|a_{nk}|<\infty$ for each $n$. Then

$\displaystyle S_{n}^{A}(g)$ $\displaystyle=\sum_{k\geq 0}a_{nk}\frac{\langle
g,\psi_{k}\rangle}{\langle e_{k},\psi_{k}\rangle}e_{k}=\sum_{k\geq
0}a_{nk}\widehat{g}(k)z^{k}\quad(g\in A^{1}),$
$\displaystyle(S_{n}^{A})^{*}(h)$ $\displaystyle=\sum_{k\geq
0}a_{nk}\frac{\langle e_{k},h\rangle}{\langle
e_{k},\psi_{k}\rangle}\psi_{k}=\sum_{k\geq
0}a_{nk}\widehat{h}(k)z^{k}\quad(h\in{\mathcal{B}}).$

It is known that $\|\sigma_{n}(g)-g\|_{A^{1}}\to 0$ for all $g\in A^{1}$ (see
Theorem 5.8 (i) below). Theorem 3.1 therefore applies. In the notation of
Theorem 3.1, $Z$ is the norm-closure in ${\mathcal{B}}$ of
$\operatorname{span}\\{z^{k}:k\geq 0\\}$, which is exactly
${\mathcal{B}}_{0}$. We thus obtain the following result.

######  Theorem 5.4.

Let $(a_{nk})_{n,k\geq 0}$ be an infinite matrix of complex scalars such that
$\sum_{k}|a_{nk}|<\infty$ for each $n$. Then the following statements are
equivalent:

  1. cmrmn(i)

$\sum_{k\geq 0}a_{nk}\widehat{g}(k)z^{k}\to g$ in $(A^{1},w)$, for all $g\in
A^{1}$;

  2. cmrmn(ii)

$\sum_{k\geq 0}a_{nk}\widehat{g}(k)z^{k}\to g$ in $(A^{1},\|\cdot\|_{A^{1}})$,
for all $g\in A^{1}$;

  3. cmrmn(iii)

$\sum_{k\geq 0}a_{nk}\widehat{h}(k)z^{k}\to h$ in $({\mathcal{B}},w^{*})$, for
all $h\in{\mathcal{B}}$;

  4. cmrmn(iv)

$\sum_{k\geq 0}a_{nk}\widehat{f}(k)z^{k}\to f$ in $({\mathcal{B}}_{0},w)$, for
all $f\in{\mathcal{B}}_{0}$;

  5. cmrmn(v)

$\sum_{k\geq 0}a_{nk}\widehat{f}(k)z^{k}\to f$ in
$({\mathcal{B}}_{0},\|\cdot\|_{\mathcal{B}})$, for all
$f\in{\mathcal{B}}_{0}$.

######  Remark.

Explicitly, statement (iii) means that, for all $h\in{\mathcal{B}}$ and all
$g\in A^{1}$,

$\lim_{n\to\infty}\lim_{r\to 1^{-}}\sum_{k\geq
0}a_{nk}r^{k}\widehat{g}(k)\widehat{h}(k)=\lim_{r\to
1^{-}}\frac{1}{\pi}\int_{|z|<r}g(z)h(\overline{z})\,dA(z).$

###  5.3. Relationship between Hardy and Bergman spaces

The following result describes the relationship between summability in $H^{p}$
and summability in $A^{p}$.

######  Theorem 5.5.

Let $1\leq p<\infty$. Let $A:=(a_{nk})_{n,k\geq 0}$ be an infinite matrix of
complex scalars such that $\sum_{k}|a_{nk}|<\infty$ for each $n$. If
$\|S_{n}^{A}(f)-f\|_{H^{p}}\to 0$ for all $f\in H^{p}$, then
$\|S_{n}^{A}(g)-g\|_{A^{p}}\to 0$ for all $g\in A^{p}$.

For the proof, we need a lemma.

######  Lemma 5.6.

Let $1\leq p<\infty$.

  1. cmrmn(1)

We have $H^{p}\subset A^{p}$, and $\|f\|_{A^{p}}\leq\|f\|_{H^{p}}$ for all
$f\in H^{p}$.

  2. cmrmn(2)

Let $S:\operatorname{Hol}({\mathbb{D}})\to\operatorname{Hol}({\mathbb{D}})$ be
a linear map such that $(Sf)_{r}=S(f_{r})$ for all
$f\in\operatorname{Hol}({\mathbb{D}})$ and all $r\in(0,1)$. If
$\|Sf\|_{H^{p}}\leq C\|f\|_{H^{p}}$ for all $f\in H^{p}$, then
$\|Sg\|_{A^{p}}\leq C\|g\|_{A^{p}}$ for all $g\in A^{p}$.

###### Proof.

(1) By Fubini’s theorem, we have the identity

$\|f\|_{A^{p}}^{p}=\int_{0}^{1}\|f_{r}\|_{H^{p}}^{p}\,2r\,dr.$

As $\|f_{r}\|_{H^{p}}\leq\|f\|_{H_{p}}~{}\forall r\in(0,1)$, we obtain
$\|f\|_{A^{p}}^{p}\leq\int_{0}^{1}\|f\|_{H^{p}}^{p}\,2r\,dr=\|f\|_{H^{p}}^{p}$.

(2) Again by the above identity, if $f\in A^{p}$, then

$\displaystyle\|Sf\|_{A^{p}}^{p}$
$\displaystyle=\int_{0}^{1}\|(Sf)_{r}\|_{H^{p}}^{p}\,2r\,dr=\int_{0}^{1}\|S(f_{r})\|_{H^{p}}^{p}\,2r\,dr$
$\displaystyle\leq
C^{p}\int_{0}^{1}\|f_{r}\|_{H^{p}}^{p}\,2r\,dr=C^{p}\|f\|_{A^{p}}^{p}.\qed$

######  Remark.

In fact a result of Hardy and Littlewood shows that we even have $H^{p}\subset
A^{2p}$. For a simple proof of this, and an example showing that $2p$ is
sharp, see the article of Vukotić [18]. However, we do not need this here.

###### Proof of Theorem 5.5.

Assume that $\|S_{n}^{A}(f)-f\|_{H^{p}}\to 0$ for all $f\in H^{p}$. By part
(1) of the lemma, $\|S_{n}^{A}(f)-f\|_{A^{p}}\to 0$ for all $f\in H^{p}$. By
part (2) of the lemma, $\|S_{n}^{A}:A^{p}\to A^{p}\|\leq\|S_{n}^{A}:H^{p}\to
H^{p}\|$ for all $n$, and by the Banach–Steinhaus theorem,
$\sup_{n}\|S_{n}^{A}:H^{p}\to H^{p}\|<\infty$. The standard density argument
now gives that $\|S_{n}^{A}(g)-g\|_{A^{p}}\to 0$ for all $g\in A^{p}$. ∎

Once again, we finish the section by applying the above work to classical
Cesàro means.

######  Theorem 5.7.

  1. cmrmn(1)

If $1<p<\infty$, then $\|s_{n}(g)-g\|_{A^{p}}\to 0$.

  2. cmrmn(2)

There exists $g\in A^{1}$ such that $s_{n}(g)\not\to g$ in $(A^{1},w)$.

  3. cmrmn(3)

There exists $f\in{\mathcal{B}}_{0}$ such that $s_{n}(f)\not\to f$ in
$({\mathcal{B}}_{0},w)$.

###### Proof.

Part (1) is a consequence of the (well-known) corresponding result in $H^{p}$
and Theorem 5.5. By a result of Zhu [19, Theorem 9], there exists $g\in A^{1}$
such that $\|s_{n}(g)-g\|_{A^{1}}\not\to 0$. Parts (2) and (3) both follow by
applying the equivalences in Theorem 5.4. ∎

######  Remark.

Zhu has previously shown that there exists $f\in{\mathcal{B}}_{0}$ such that
$\|s_{n}(f)-f\|_{{\mathcal{B}}}\not\to 0$ (see [19, Corollary 11]). The same
result had also been obtained earlier by Anderson, Clunie and Pommerenke [2],
but with a slightly different identification of the dual of
${\mathcal{B}}_{0}$ and the predual of ${\mathcal{B}}$.

######  Theorem 5.8.

Let $\alpha>0$.

  1. cmrmn(1)

$\|\sigma_{n}^{\alpha}(g)-g\|_{A^{1}}\to 0$ for all $g\in A^{1}$.

  2. cmrmn(2)

$\sigma_{n}^{\alpha}(h)\to h$ in $({\mathcal{B}},w^{*})$ for all
$h\in{\mathcal{B}}$.

  3. cmrmn(3)

$\|\sigma_{n}^{\alpha}(f)-f\|_{{\mathcal{B}}}\to 0$ for all
$f\in{\mathcal{B}}_{0}$.

###### Proof.

Part (1) follows from the corresponding result for $H^{1}$, together with
Theorem 5.5. Parts (2) and (3) are consequences of (1), using the equivalences
in Theorem 5.4. ∎

##  6\. Applications in Hilbert spaces

###  6.1. Abstract set-up

In the case of a Hilbert space, we can repeat the analysis of §3 using the
inner product in place of the duality pairing. The fact that an inner product
is sesquilinear rather than bilinear leads to some slight differences between
the two theories.

Throughout this section, $H$ denotes a complex Hilbert space with inner
product $\langle\cdot,\cdot\rangle$. Also, if $T$ is a bounded linear operator
on $H$, then $T^{*}$ denotes the Hilbert-space adjoint of $T$, namely the
unique operator on $H$ such that

$\langle Tg,h\rangle=\langle g,T^{*}h\rangle\quad(g,h\in H).$

The following theorem is the analogue of Theorem 2.1. It is proved in just the
same way, so we omit the details. Of course, since a Hilbert space is
reflexive, there is no distinction between weak- and weak*-convergence.

######  Theorem 6.1.

Let $(T_{n})_{n\geq 1}$ be a sequence of bounded, finite-rank operators on $H$
such that

$T_{n}T_{m}(H)\subset T_{m}(H)\quad\text{and}\quad
T_{n}^{*}T_{m}^{*}(H)\subset T_{m}^{*}(H)\qquad(m,n\geq 0).$

Then the following statements are equivalent:

  1. cmrmn(i)

$T_{n}h\to h$ weakly for all $h\in H$;

  2. cmrmn(ii)

$T_{n}h\to h$ in norm for all $h\in H$;

  3. cmrmn(iii)

$T_{n}^{*}h\to h$ weakly for all $h\in H$;

  4. cmrmn(iv)

$T_{n}^{*}h\to h$ in norm for all $h\in H$.

Now suppose that $(e_{k})_{k\geq 0}$ and $(f_{k})_{k\geq 0}$ are two sequences
in $H$ such that

$\left\\{\begin{aligned} \langle e_{j},f_{k}\rangle&=0,\quad\forall
j,k,~{}j\neq k,\\\ \langle e_{k},f_{k}\rangle&\neq 0,\quad\forall
k.\end{aligned}\right.$

For each $k\geq 0$, define $P_{k}:H\to H$ by

$P_{k}:=\frac{e_{k}\otimes f_{k}}{\langle e_{k},f_{k}\rangle}.$

Explicitly,

$P_{k}(h):=\frac{\langle h,f_{k}\rangle}{\langle
e_{k},f_{k}\rangle}e_{k}\quad(h\in H).$

Clearly $P_{k}(e_{k})=e_{k}$ and $P_{k}(e_{j})=0$ if $j\neq k$. It is easy to
see that $P_{k}$ is a rank-one projection with

$\|P_{k}\|=\frac{\|e_{k}\|\|f_{k}\|}{|\langle e_{k},f_{k}\rangle|}.$

Its adjoint $P_{k}^{*}:H\to H$ is given by

$P_{k}^{*}=\frac{f_{k}\otimes e_{k}}{\langle f_{k},e_{k}\rangle}.$

Let $A=(a_{nk})_{n,k\geq 0}$ be an infinite matrix of complex scalars such
that, for each $n\geq 0$,

$\sum_{k\geq 0}|a_{nk}|\|P_{k}\|<\infty.$

We define the associated summation operators $S_{n}^{A}:H\to H~{}(n\geq 0)$ by
the absolutely convergent series

$S_{n}^{A}:=\sum_{k\geq 0}a_{nk}P_{k}.$

Explicitly, we have

$\displaystyle S_{n}^{A}(h)$ $\displaystyle=\sum_{k\geq 0}a_{nk}\frac{\langle
h,f_{k}\rangle}{\langle e_{k},f_{k}\rangle}e_{k}\quad(h\in H),$
$\displaystyle(S_{n}^{A})^{*}(h)$ $\displaystyle=\sum_{k\geq
0}\overline{a}_{nk}\frac{\langle h,e_{k}\rangle}{\langle
f_{k},e_{k}\rangle}f_{k}\quad(h\in H).$

With this notation established, we have the following equivalence theorem. It
follows from Theorem 6.1 in just the same way that Theorem 3.1 follows from
Theorem 2.1. We omit the details.

######  Theorem 6.2.

The following statements are equivalent:

  1. cmrmn(i)

$S_{n}^{A}(h)\to h$ weakly, for all $h\in H$.

  2. cmrmn(ii)

$S_{n}^{A}(h)\to h$ in norm, for all $h\in H$.

  3. cmrmn(iii)

$(S_{n}^{A})^{*}(h)\to h$ weakly, for all $h\in H$.

  4. cmrmn(iv)

$(S_{n}^{A})^{*}(h)\to h$ in norm, for all $h\in H$.

There are also Hilbert-space versions of the limitation theorems, Theorem 3.2
and Theorem 3.4. The proofs are the same as before.

######  Theorem 6.3.

Suppose that $S_{n}^{A}(x)\to x$ (weakly or in norm) for all $x\in H$, and
that $A$ admits a left inverse $B$. Then, writing $B_{j}:=\sum_{n\geq
0}|b_{jn}|$, we have

$\frac{\|e_{j}\|\|f_{j}\|}{|\langle
e_{j},f_{j}\rangle|}=O(B_{j})\quad(j\to\infty).$

######  Theorem 6.4.

Let $\alpha\geq 0$. If $\sigma_{n}^{\alpha}(x)\to x$ (weakly or in norm) for
all $x\in H$, then

$\frac{\|e_{j}\|\|f_{j}\|}{|\langle
e_{j},f_{j}\rangle|}=O(j^{\alpha})\quad(j\to\infty).$

###  6.2. Reproducing kernel spaces of holomorphic functions

We shall now apply these ideas to reproducing kernel spaces of holomorphic
functions on the unit disk.

Let $H$ be a Hilbert space of holomorphic functions on ${\mathbb{D}}$ such
that:

  * •

convergence in the norm of $H$ implies pointwise convergence on
${\mathbb{D}}$;

  * •

$H$ contains the polynomials.

The first assumption implies that, for each $w\in{\mathbb{D}}$, the functional
$h\mapsto h(w)$ is continuous on $H$, so, by the Riesz representation theorem,
there exists a unique $k_{w}\in H$ such that

$h(w)=\langle h,k_{w}\rangle\quad(h\in H).$

We define $K:H\times H\to{\mathbb{C}}$ by

$K(z,w):=k_{w}(z)=\langle k_{w},k_{z}\rangle\quad(z,w\in{\mathbb{D}}).$

The function $K$ is the reproducing kernel of $H$. Clearly it satisfies
$K(w,z)=\overline{K(z,w)}$, and $K(z,w)$ is holomorphic in $z$ for each fixed
$w$. Therefore it is anti-holomorphic in $w$ for each fixed $z$. We shall need
the following simple lemma about derivatives of $K$.

######  Lemma 6.5.

For each $w\in{\mathbb{D}}$ and each $n\geq 0$, let

$k_{w,n}(z):=\frac{1}{n!}\frac{\partial^{n}}{\partial\overline{w}^{n}}K(z,w)\quad(z\in{\mathbb{D}}).$

Then $k_{w,n}\in H$ and

(6.1) $\langle h,k_{w,n}\rangle=\frac{h^{(n)}(w)}{n!}\quad(h\in H).$

###### Proof.

As norm-convergence in $H$ implies local uniform convergence on
${\mathbb{D}}$, the map $h\mapsto h^{(n)}(w)/n!$ is a continuous linear
functional on $H$, so, by the Riesz theorem again, there exists $k_{w,n}\in H$
such that (6.1) holds. It remains to identify $k_{w,n}$, which we do as
follows. For each $z\in{\mathbb{D}}$, we have

$\displaystyle k_{w,n}(z)$ $\displaystyle=\langle
k_{w,n},k_{z}\rangle=\overline{\langle k_{z},k_{w,n}\rangle}$
$\displaystyle=\overline{\frac{k_{z}^{(n)}(w)}{n!}}=\overline{\frac{1}{n!}\frac{\partial^{n}}{\partial
w^{n}}K(w,z)}=\frac{1}{n!}\frac{\partial^{n}}{\partial\overline{w}^{n}}K(z,w).\qed$

Now we define sequences $(e_{j})_{j\geq 0}$ and $(f_{j})_{j\geq 0}$ in $H$ by
$e_{j}:=z^{j}$ and $f_{j}:=k_{0,j}$. Note that, if $h\in H$, then

$\langle h,f_{j}\rangle=\frac{h^{(j)}(0)}{j!}=\widehat{h}(j).$

In particular, $\langle e_{i},f_{j}\rangle=\delta_{ij}$, so the theory
outlined in §6.1 applies. Note also that $\|e_{j}\|=\|z^{j}\|_{H}$ and

$\|f_{j}\|^{2}=\|k_{0,j}\|^{2}=\langle
k_{0,j},k_{0,j}\rangle=\frac{1}{j!}k_{0,j}^{(j)}(0)=\frac{1}{j!^{2}}\frac{\partial^{2j}K}{\partial
z^{j}\partial\overline{w}^{j}}(0,0).$

Theorem 6.2 leads to the following result.

######  Theorem 6.6.

Let $(a_{nj})_{n,j\geq 0}$ be an infinite matrix of complex scalars such that
$\sum_{j\geq 0}|a_{nj}|\|z^{j}\|\|k_{0,j}\|<\infty$ for each $n\geq 0$. Then
the following statements are equivalent:

  1. cmrmn(i)

$\sum_{j\geq 0}a_{nj}\widehat{h}(j)z^{j}\to h$ weakly as $n\to\infty$, for all
$h\in H$;

  2. cmrmn(ii)

$\sum_{j\geq 0}a_{nj}\widehat{h}(j)z^{j}\to h$ in norm as $n\to\infty$, for
all $h\in H$;

  3. cmrmn(iii)

$\sum_{j\geq 0}\overline{a}_{nj}\langle h,z^{j}\rangle k_{0,j}\to h$ weakly as
$n\to\infty$, for all $h\in H$;

  4. cmrmn(iv)

$\sum_{j\geq 0}\overline{a}_{nj}\langle h,z^{j}\rangle k_{0,j}\to h$ in norm
as $n\to\infty$, for all $h\in H$.

The following result is a limitation theorem for reproducing kernel spaces. It
is an immediate consequence of Theorem 6.4.

######  Theorem 6.7.

Let $H$ be a reproducing-kernel space of holomorphic functions on
${\mathbb{D}}$ that contains the polynomials. If $\alpha\geq 0$ and
$\sigma_{n}^{\alpha}(h)\to h$ (weakly or in norm) for all $h\in H$, then

$\|z^{j}\|\|k_{0,j}\|=O(j^{\alpha})\quad(j\to\infty).$

###  6.3. De Branges–Rovnyak spaces

We now specialize to the case where $H={\mathcal{H}}(b)$, the de
Branges–Rovnyak space with symbol $b$. Here $b$ is an element of the unit ball
of $H^{\infty}$. By definition, ${\mathcal{H}}(b)$ is the reproducing-kernel
space on ${\mathbb{D}}$ with kernel

$K(z,w)=\frac{1-b(z)\overline{b(w)}}{1-z\overline{w}}\quad(z,w\in{\mathbb{D}}).$

The space ${\mathcal{H}}(b)$ contains the polynomials iff $b$ is a non-extreme
point of the unit ball of $H^{\infty}$, and in this case the polynomials are
dense in ${\mathcal{H}}(b)$. It is known that $b$ is non-extreme iff its
boundary values satisfy $\log(1-|b|^{2})\in L^{1}({\mathbb{T}})$. Henceforth,
we assume that this is the case. For further information on
${\mathcal{H}}(b)$-spaces, we refer to [16] and [4, 5].

The following result will be useful in what follows.

######  Proposition 6.8.

Let $b$ be a non-extreme point of the unit ball of $H^{\infty}$. Then
$\inf_{j\geq 0}\|k_{0,j}\|_{{\mathcal{H}}(b)}>0$.

###### Proof.

We compute an expression for $k_{0,j}$ in ${\mathcal{H}}(b)$. Using Leibniz’
theorem, we have

$\displaystyle k_{0,j}(z)$
$\displaystyle=\frac{1}{j!}\frac{\partial^{j}}{\partial\overline{w}^{j}}K(z,0)$
$\displaystyle=\frac{1}{j!}\sum_{i=0}^{j}\binom{j}{i}\frac{\partial^{i}}{\partial\overline{w}^{i}}\Bigl{(}1-b(z)\overline{b(w)}\Bigr{)}\Bigl{|}_{w=0}\frac{\partial^{j-i}}{\partial\overline{w}^{j-i}}\Bigl{(}1-z\overline{w}\Bigr{)}^{-1}\Bigl{|}_{w=0}$
$\displaystyle=z^{j}-b(z)\sum_{i=0}^{j}\overline{\widehat{b}(i)}z^{j-i}.$

Thus, viewed as functions on ${\mathbb{T}}$,

$k_{0,j}=z^{j}(1-b\overline{s_{j}(b)}).$

Also, expanding $K(z,w)=(1-b(z)\overline{b(w)})/(1-z\overline{w})$ as a double
power series in $z,\overline{w}$, and computing the coefficient of
$z^{j}\overline{w}^{j}$, we find that

$\|k_{0,j}\|_{{\mathcal{H}}(b)}^{2}=\frac{1}{j!^{2}}\frac{\partial^{2j}K}{\partial
z^{j}\partial\overline{w}^{j}}(0,0)=1-\sum_{i=0}^{j}|\widehat{b}(i)|^{2}=1-\|s_{j}(b)\|_{H^{2}}^{2}.$

In particular, we have

$1\geq\|k_{0,j}\|_{{\mathcal{H}}(b)}^{2}\geq 1-\|b\|_{H^{2}}^{2}\quad(j\geq
0).$

As $b$ is non-extreme, $\|b\|_{H^{2}}<\|b\|_{H^{\infty}}\leq 1$ or
$\|b\|_{H^{2}}=\|b\|_{H^{\infty}}<1$. Either way, $1-\|b\|_{H^{2}}^{2}>0$.
Hence $\inf_{j\geq 0}\|k_{0,j}\|_{{\mathcal{H}}(b)}>0$, as claimed. ∎

######  Remark.

A similar result holds in the more general finite-rank
${\mathcal{H}}[B]$-spaces studied by Aleman and Malman in [1]. However, not
all reproducing kernel function spaces have this property. For example, in the
classical Dirichlet space on the unit disk, we have $\|k_{0,j}\|\asymp 1/j$.

Feeding this information into the results of §6.2, we deduce the following
results. (We write $\langle\cdot,\cdot\rangle_{b}$ for the inner product in
${\mathcal{H}}(b)$.)

######  Theorem 6.9.

Let $b$ be a non-extreme point of the unit ball of $H^{\infty}$. Let
$(a_{nj})_{n,j\geq 0}$ be a matrix of complex scalars such that $\sum_{j\geq
0}|a_{nj}|\|z^{j}\|_{{\mathcal{H}}(b)}<\infty$ for each $n\geq 0$. Then the
following statements are equivalent:

  1. cmrmn(i)

$\sum_{j\geq 0}a_{nj}\widehat{h}(j)z^{j}\to h$ weakly in ${\mathcal{H}}(b)$,
for all $h\in{\mathcal{H}}(b)$;

  2. cmrmn(ii)

$\sum_{j\geq 0}a_{nj}\widehat{h}(j)z^{j}\to h$ in norm in ${\mathcal{H}}(b)$,
for all $h\in{\mathcal{H}}(b)$;

  3. cmrmn(iii)

$\sum_{j\geq 0}\overline{a}_{nj}(1-b\overline{s_{j}(b)})\langle
h,z^{j}\rangle_{b}z^{j}\to h$ weakly in ${\mathcal{H}}(b)$, for all
$h\in{\mathcal{H}}(b)$;

  4. cmrmn(iv)

$\sum_{j\geq 0}\overline{a}_{nj}(1-b\overline{s_{j}(b)})\langle
h,z^{j}\rangle_{b}z^{j}\to h$ in norm in ${\mathcal{H}}(b)$, for all
$h\in{\mathcal{H}}(b)$.

######  Theorem 6.10.

Let $b$ be a non-extreme point of the unit ball of $H^{\infty}$. If
$\alpha\geq 0$ and $\sigma_{n}^{\alpha}(h)\to h$ (weakly or in norm) for all
$h\in{\mathcal{H}}(b)$, then

$\|z^{j}\|_{{\mathcal{H}}(b)}=O(j^{\alpha})\quad(j\to\infty).$

The following consequence is worth pointing out explicitly.

######  Corollary 6.11.

Let $b$ be a non-extreme point of the unit ball of $H^{\infty}$. If
$s_{n}(h)\to h$ (weakly or in norm) for all $h\in{\mathcal{H}}(b)$, then
$\sup_{j\geq 0}\|z^{j}\|_{{\mathcal{H}}(b)}<\infty$.

The de Branges–Rovnyak spaces in which the powers of $z$ are bounded in norm
form an interesting class in their own right. For example, they are precisely
the ${\mathcal{H}}(b)$-spaces that contain $H^{\infty}$. Also, they are
characterized by the condition that $1/(1-|b|^{2})\in L^{1}({\mathbb{T}})$.
For more on this, see [15, §4].

For $\alpha>0$, the convergence of $\sigma_{n}^{\alpha}(h)$ to $h$ for every
$h\in{\mathcal{H}}(b)$ also has implications for $b$. To derive these, we need
an explicit formula for $\|z^{j}\|_{{\mathcal{H}}(b)}$, which requires that we
delve a little further into the theory of de Branges–Rovnyak spaces. The
details can be found in [16] and [5].

As mentioned earlier, if $b$ is a non-extreme point of the unit ball of
$H^{\infty}$, then $\log(1-|b|^{2})\in L^{1}({\mathbb{T}})$. This implies that
there exists a unique outer function $a$ on ${\mathbb{D}}$ with $a(0)>0$ such
that $|b|^{2}+|a|^{2}=1$ a.e. on ${\mathbb{T}}$. This function $a$ is
sometimes called the pythagorean complement of $b$. Writing $\phi:=b/a$, we
obtain a function in the Smirnov class $N^{+}$, namely the space of quotients
of $H^{\infty}$-functions with outer denominators. Conversely, all Smirnov
functions are obtained in this way. There is thus a one-to-one correspondence
$b\leftrightarrow\phi$, between non-extreme points $b$ of the unit ball of
$H^{\infty}$ and functions $\phi\in N^{+}$. Expanding $\phi(z)$ as a Taylor
series, say $\phi(z)=\sum_{j\geq 0}c_{j}z^{j}$, we have

(6.2) $\|z^{j}\|_{{\mathcal{H}}(b)}^{2}=1+\sum_{i=0}^{j}|c_{i}|^{2}.$

A simple proof of (6.2) can be found for example in [15, p.81].

######  Theorem 6.12.

Let $b$ be a non-extreme point of the unit ball of $H^{\infty}$ and let $\phi$
be the corresponding function in the Smirnov class. If $\alpha\geq 0$ and
$\sigma_{n}^{\alpha}(h)\to h$ (weakly or in norm) for all
$h\in{\mathcal{H}}(b)$, then

(6.3) $\phi(z)=O\Bigl{(}(1-|z|)^{-\alpha-1/2}\Bigr{)}\quad(|z|\to 1^{-}).$

###### Proof.

As above, let us write $\phi(z)=\sum_{j\geq 0}c_{j}z^{j}$. Summing by parts,
we have

$|\phi(z)|\leq\sum_{j\geq 0}|c_{j}||z|^{j}=(1-|z|)\sum_{j\geq
0}\Bigl{(}\sum_{i=0}^{j}|c_{i}|\Bigr{)}|z|^{j}\quad(|z|<1).$

By Cauchy–Schwarz and formula (6.2),

$\sum_{i=0}^{j}|c_{i}|\leq(j+1)^{1/2}\Bigl{(}\sum_{i=0}^{j}|c_{i}|^{2}\Bigr{)}^{1/2}\leq(j+1)^{1/2}\|z^{j}\|_{{\mathcal{H}}(b)}\quad(j\geq
0).$

From Theorem 6.10, it follows that

$\sum_{i=0}^{j}|c_{i}|=O(j^{\alpha+1/2})\quad(j\to\infty).$

Feeding this information back into the inequality for $\phi$, we find that
there is a constant $C$ such that

$|\phi(z)|\leq C(1-|z|)\sum_{j\geq 0}j^{\alpha+1/2}|z|^{j}\quad(|z|<1).$

The conclusion (6.3) follows. ∎

We conclude with two concrete examples.

######  Corollary 6.13.

Let $\phi(z):=z/(1-z)$ and let $b$ be the corresponding non-extreme point of
the unit ball of $H^{\infty}$. If $\sigma^{\alpha}_{n}(h)\to h$ for all
$h\in{\mathcal{H}}(b)$, then $\alpha\geq 1/2$.

######  Remark.

In this case $b$ can be determined explicitly, namely as

$b(z)=\frac{(1-w_{0})z}{1-w_{0}z},$

where $w_{0}:=(3-\sqrt{5})/2$. This example is taken from [17], where it is
shown that, with this particular choice of $b$, the de Branges–Rovynak space
${\mathcal{H}}(b)$ is equal to the local Dirichlet space
${\mathcal{D}}(\delta_{1})$ (with the same norm). It was shown in [9] that, if
$\alpha>1/2$, then $\sigma_{n}^{\alpha}(h)\to h$ in norm for all
$h\in{\mathcal{D}}(\delta_{1})$, but that there exists
$h_{0}\in{\mathcal{D}}(\delta_{1})$ such that $\sigma_{n}^{1/2}(h_{0})\not\to
h_{0}$. Thus the constant $1/2$ in Corollary 6.13 is sharp.

######  Corollary 6.14.

Let

$\phi(z):=\exp\Bigl{(}\sqrt{\frac{1+z}{1-z}}\Bigr{)}\quad(z\in{\mathbb{D}}).$

Then $\phi\in N^{+}$. If $b$ is the corresponding non-extreme point of the
unit ball of $H^{\infty}$, then there exists no $\alpha\geq 0$ such that
$\sigma_{n}^{\alpha}(h)\to h$ for all $h\in{\mathcal{H}}(b)$.

###### Proof.

The function $\sqrt{(1+z)/(1-z)}$ belongs to $H^{1}$. Its exponential is
therefore an outer function, so $\phi\in N^{+}$. Now apply Theorem 6.12. ∎

######  Remark.

Thus, in this case, even though polynomials are dense in ${\mathcal{H}}(b)$,
there is no Cesàro summability method that always converges. Such examples
were previously obtained in [3] and [12] using inductive constructions. Our
method has the virtue that it yields a (fairly simple) explicit function
$\phi$ that does the trick.

### Acknowledgements

We thank Ryan Gibara for helpful discussions.

## References

  * [1] A. Aleman and B. Malman.  Hilbert spaces of analytic functions with a contractive backward shift.  J. Funct. Anal., 277(1):157–199, 2019.
  * [2] J. M. Anderson, J. Clunie, and Ch. Pommerenke.  On Bloch functions and normal functions.  J. Reine Angew. Math., 270:12–37, 1974.
  * [3] O. El-Fallah, E. Fricain, K. Kellay, J. Mashreghi, and T. Ransford.  Constructive approximation in de Branges-Rovnyak spaces.  Constr. Approx., 44(2):269–281, 2016.
  * [4] E. Fricain and J. Mashreghi.  The theory of $\mathcal{H}$($b$) spaces. Vol. 1, volume 20 of New Mathematical Monographs.  Cambridge University Press, Cambridge, 2016.
  * [5] E. Fricain and J. Mashreghi.  The theory of $\mathcal{H}(b)$ spaces. Vol. 2, volume 21 of New Mathematical Monographs.  Cambridge University Press, Cambridge, 2016.
  * [6] J. B. Garnett.  Bounded analytic functions, volume 236 of Graduate Texts in Mathematics.  Springer, New York, reprint of first edition, 2007.
  * [7] G. H. Hardy.  On the Summability of Fourier’s Series.  Proc. London Math. Soc. (2), 12:365–372, 1913.
  * [8] G. H. Hardy.  Divergent series.  Chelsea, New York, reprint of second edition, 1991.
  * [9] J. Mashreghi, P.-O. Parisé, and T. Ransford.  Cesàro summability of Taylor series in weighted Dirichlet spaces.  Complex Anal. Oper. Theory, 15(1):Paper No. 7, 8, 2021.
  * [10] J. Mashreghi, P.-O. Parisé, and T. Ransford.  Failure of Approximation of Odd Functions by Odd Polynomials.  Constr. Approx., 56(1):35–43, 2022.
  * [11] J. Mashreghi, P.-O. Parisé, and T. Ransford.  Power-series summability methods in de Branges–Rovnyak spaces.  Integral Equations Operator Theory, 94(2):Paper No. 20, 17, 2022\.
  * [12] J. Mashreghi and T. Ransford.  Outer functions and divergence in de Branges–Rovnyak spaces.  Complex Anal. Oper. Theory, 12(4):987–995, 2018.
  * [13] J. Mashreghi and T. Ransford.  Linear polynomial approximation schemes in Banach holomorphic function spaces.  Anal. Math. Phys., 9(2):899–905, 2019.
  * [14] M. Riesz.  Sur les séries de Dirichlet et les séries entières,.  C. R. Acad. Sci. Paris, 149:309–312, 1911.
  * [15] D. Sarason.  Doubly shift-invariant spaces in $H^{2}$.  J. Operator Theory, 16(1):75–97, 1986.
  * [16] D. Sarason.  Sub-Hardy Hilbert spaces in the unit disk, volume 10 of University of Arkansas Lecture Notes in the Mathematical Sciences.  John Wiley & Sons, Inc., New York, 1994.  A Wiley-Interscience Publication.
  * [17] D. Sarason.  Local Dirichlet spaces as de Branges-Rovnyak spaces.  Proc. Amer. Math. Soc., 125(7):2133–2139, 1997.
  * [18] D. Vukotić.  The isoperimetric inequality and a theorem of Hardy and Littlewood.  Amer. Math. Monthly, 110(6):532–536, 2003.
  * [19] K. Zhu.  Duality of Bloch spaces and norm convergence of Taylor series.  Michigan Math. J., 38(1):89–101, 1991.
  * [20] K. Zhu.  Operator theory in function spaces, volume 138 of Mathematical Surveys and Monographs.  American Mathematical Society, Providence, RI, second edition, 2007.