Approximation numbers of composition operators on the Hardy space of the infinite polydisk

TL;DR

This paper investigates the approximation numbers of composition operators acting on the Hardy space of the infinite polydisk, focusing on their behavior within the  part of the space.

Contribution

It provides new insights into the approximation numbers of composition operators on Hardy spaces of the infinite polydisk, a less-explored area in functional analysis.

Findings

Characterization of approximation numbers for specific composition operators

Analysis of the decay rates of approximation numbers in this setting

New bounds or estimates for these approximation numbers

Abstract

We study the composition operators of the Hardy space on D $\infty$ $\cap$ {\ell} 1 , the {\ell} 1 part of the infinite polydisk, and the behavior of their approximation numbers.

Equations

$$a_n(C_{\phi })\le C{\mathrm{e}}^{-c{n}^{1/d}}.$$

$$\hat{f}(\alpha ):={\int }_{{\mathbb{T}}^{\infty }}f(z){\overline{z}}^{\alpha }dm(z)=0\text{if }\alpha \notin {\mathbb{N}}^{(\infty )}.$$

$$f=\sum _{\alpha \ge 0}{c}_{\alpha }{e}_{\alpha }\text{with }{c}_{\alpha }=\hat{f}(\alpha )\text{and}\sum _{\alpha \ge 0}|c_{\alpha }{|}^2<\infty .$$

$$f(z)=\sum _{\alpha \ge 0}{c}_{\alpha }{z}^{\alpha }$$

$$\sum _{\alpha \ge 0}|z^{\alpha }{|}^2=\prod _{j=1}^{\infty }(1-|z_j{|}^2{)}^{-1}<\infty ,$$

$$\sum_{\alpha\geq 0}|c_{\alpha}\,z^{\alpha}|\leq\bigg{(}\sum_{\alpha\geq 0}|c_{\alpha}|^{2}\bigg{)}^{1/2}\bigg{(}\sum_{\alpha\geq 0}|z^{\alpha}|^{2}\bigg{)}^{1/2}<\infty\,.$$

$$\Vert f{\Vert }^2:=\sum _{\alpha \ge 0}|c_{\alpha }{|}^2<\infty .$$

$$\Omega ={\mathbb{D}}^{\infty }\cap {\ell }_1,$$

$$z=(z_n{)}_{n\ge 1}\text{such that}|z_n|<1,\forall n\ge 1,\text{and}\sum _{n=1}^{\infty }|z_n|<\infty ,$$

$$f(z)=\sum _{\alpha \ge 0}{c}_{\alpha }{z}^{\alpha }\text{with }z\in \Omega ,$$

$$H^p=H^p(\Omega )=\{f:\Omega \to \mathbb{C};\Vert f{\Vert }_p<\infty \},$$

$$M_{p}(r,f)=\bigg{(}\int_{\mathbb{T}^{\infty}}|f(rz)|^{p}\,dm(z)\bigg{)}^{1/p},\quad 0<r<1\,.$$

$$a_n=\underset{\text{rank}R<n}{\inf }\Vert T-R\Vert .$$

$$\prod _{j=1}^n|{\lambda }_j|\le \prod _{j=1}^n{a}_j\text{for all }n\ge 1.$$

$$|{\lambda }_{2n}{|}^2\le a_1{a}_n.$$

$$T({\otimes }_{i=1}^n{x}_i)={\otimes }_{i=1}^n{T}_i(x_i).$$

$$\sigma (x_1\cdots x_n)\subseteq \prod _{i=1}^n\sigma (x_i).$$

$$\prod_{i=1}^{n}\sigma(x_{i})=\bigg{\{}\prod_{i=1}^{n}\lambda_{i}:\lambda_{i}\in\sigma(x_{i})\bigg{\}}\,.$$

$$x_1{x}_2=x_2{x}_1=T_1\otimes T_2=T\text{and}\sigma (x_i)=\sigma (T_i)$$

$$\sigma (T)=\sigma (x_1{x}_2)\subseteq \sigma (x_1)\sigma (x_2)=\sigma (T_1)\sigma (T_2),$$

$$|{\phi }_j(z_j)-{\phi }_j(0)|\le 2|z_j|$$

$$\Vert C_{{\phi }_j}\Vert \le \sqrt{\frac{1+|{\phi }_j(0)|}{1-|{\phi }_j(0)|}}\cdot$$

$$\Vert C_{\phi }\Vert \le \prod _{j=1}^{\infty }\Vert C_{{\phi }_j}\Vert <\infty .$$

$$\prod _{j\ge 1}\sqrt{\frac{1+|{\phi }_j(0)|}{1-|{\phi }_j(0)|}}$$

$$\Psi_{a}(z)=\big{(}\Phi_{a_{j}}(z_{j})\big{)}_{j\geq 1}$$

$$z,w\in U\text{and}f\in A⟹|f(z)-f(w)|\le C_{A,U}\Vert z-w\Vert .$$

$$\varphi(z)=\sum_{|\alpha|=1}c_{\alpha}z^{\alpha}+\sum_{s=2}^{\infty}\bigg{(}\sum_{|\alpha|=s}c_{\alpha}z^{\alpha}\bigg{)}=A(z)+\sum_{s=2}^{\infty}\bigg{(}\sum_{|\alpha|=s}c_{\alpha}z^{\alpha}\bigg{)}\,,$$

$$\Vert z{\Vert }_1<1⟹z\in \Omega ⟹A(z)=\frac{1}{2\pi }{\int }_0^{2\pi }\phi (z{\mathrm{e}}^{i\theta }){\mathrm{e}}^{-i\theta }d\theta .$$

$${\lambda }^{\alpha }=\prod _{j\ge 1}{\lambda }_j^{{\alpha }_j}\text{if}\alpha =({\alpha }_j{)}_{j\ge 1}\in {\mathbb{N}}^{(\infty )}.$$

$$f(z)=\sum _{m=0}^{\infty }\frac{d^{m}f(0)}{m!}(z^m)$$

Full text

Approximation numbers of composition operators on the Hardy space of the infinite polydisk

Daniel Li, Hervé Queffélec, L. Rodríguez-Piazza

Abstract. We study the composition operators of the Hardy space on $\mathbb{D}^{\infty}\cap\ell_{1}$, the $\ell_{1}$ part of the infinite polydisk, and the behavior of their approximation numbers.

1 Introduction

Recently, in [2], we investigated approximation numbers $a_{n}(C_{\varphi}),n\geq 1$, of composition operators $C_{\varphi},\ C_{\varphi}(f)=f\circ\varphi$, on the Hardy or Bergman spaces $H^{2}(\Omega)$, $B^{2}(\Omega)$ over a bounded symmetric domain $\Omega\subseteq\mathbb{C}^{d}$. Assuming that $\varphi(\Omega)$ has non-empty interior, one of the main results of this study was the following theorem.

Theorem 1.1** ([2]).**

Let $C_{\varphi}\colon H^{2}(\Omega)\to H^{2}(\Omega)$ be compact. Then:

$1)$* we always have $a_{n}(C_{\varphi})\geq c\,{\rm e}^{-C\,n^{1/d}}$ where $c,C$ are positive constants;*

$2)$* if $\Omega$ is a product of balls and if $\varphi(\Omega)\subseteq r\,\Omega$ for some $r<1$, then:*

[TABLE]

As a result, the minimal decay of approximation numbers is slower and slower as the dimension $d$ increases, which might lead one to think that, in infinite-dimension, no compact composition operators can exist, since their approximation numbers will not tend to [math]. After all, this is the case for the Hardy space of a half-plane, which supports no compact composition operator ([12], Theorem 3.1; in [9], it is moreover proved that $\|C_{\varphi}\|_{e}=\|C_{\varphi}\|$ as soon as $C_{\varphi}$ is bounded; see also [15] for a necessary and sufficient condition for $H^{2}(\Omega)$ has compact composition operators, where $\Omega$ is a domain of $\mathbb{C}$). We will see that this is not quite the case here, even though the decay will be severely limited. In particular, we will never have a decay of the form $C\,{\rm e}^{-c\,n^{\delta}}$ for some $c,C,\delta>0$.

2 Framework and reminders

2.1 Hardy spaces on $\mathbb{D}^{\infty}$

Let $\mathbb{T}=\partial\mathbb{D}$ be the unit circle of the set of complex numbers. We consider $\mathbb{T}^{\infty}$ and equip it with its Haar measure $m$. This is a compact Abelian group with dual $\mathbb{Z}^{(\infty)}$, the set of eventually zero sequences $\alpha=(\alpha_{j})_{j\geq 1}$ of integers. We denote $L^{2}_{\mathbb{N}^{(\infty)}}(\mathbb{T}^{\infty})$ the Hilbert subspace of $L^{2}(\mathbb{T}^{\infty})$ formed by the functions $f$ whose Fourier spectrum is contained in $\mathbb{N}^{(\infty)}$:

[TABLE]

The set $E:=\mathbb{N}^{(\infty)}$ is called the narrow cone of Helson, and we also denote $L^{2}_{\mathbb{N}^{(\infty)}}(\mathbb{T}^{\infty})=L^{2}_{E}(\mathbb{T}^{\infty})$. Any element of that subspace can be formally written as:

[TABLE]

Here, $(e_{\alpha})_{\alpha\in\mathbb{Z}^{(\infty)}}$ is the canonical basis of $L^{2}(\mathbb{T}^{\infty})$ formed by characters, and accordingly $(e_{\alpha})_{\alpha\in\mathbb{N}^{(\infty)}}$ is the canonical basis of $L^{2}_{E}(\mathbb{T}^{\infty})$.

Now we consider $\Omega_{2}=\mathbb{D}^{\infty}\cap\ell_{2}$.

Any $f\sim\sum_{\alpha\geq 0}c_{\alpha}\,e_{\alpha}\in L^{2}_{E}(\mathbb{T}^{\infty})$ defines an analytic function on the infinite-dimensional Reinhardt domain $\Omega_{2}$ by the formula:

[TABLE]

where the series is absolutely convergent for each $z=(z_{j})_{j\geq 1}\in\Omega_{2}$, as the pointwise product of two square-summable sequences. Indeed, using an Euler type formula, we get for $z\in\Omega_{2}$:

[TABLE]

and hence, by the Cauchy-Schwarz inequality:

[TABLE]

If $\alpha\in E$ and $z\in\Omega_{2}$, we have set, as usual, $z^{\alpha}=\prod_{j\geq 1}z_{j}^{\alpha_{j}}$.

This shows that $L^{2}_{E}(\mathbb{T}^{\infty})$ can be identified with $H^{2}(\Omega_{2})$, the Hardy-Hilbert space of analytic functions $f(z)=\sum_{\alpha\geq 0}c_{\alpha}\,z^{\alpha}$ on $\Omega_{2}$ with

[TABLE]

This setting is customary in connection with Dirichlet series (see [7]).

In this paper, for a technical reason appearing below in the proof of Proposition 2.5, we will consider, instead of $\Omega_{2}=\mathbb{D}^{\infty}\cap\ell_{2}$, the sub-domain:

[TABLE]

i.e. the open subset of $\ell^{1}$ formed by the sequences:

[TABLE]

and the restrictions to $\Omega$ of the functions $f\in H^{2}(\Omega_{2})$. We denote $H^{2}(\Omega)$ the space of such restrictions.

Hence $f\in H^{2}(\Omega)$ if and only if:

[TABLE]

and $\|f\|^{2}:=\sum_{\alpha\geq 0}|c_{\alpha}|^{2}<\infty$.

We now identify the space $L^{2}_{E}(\mathbb{T}^{\infty})$ with the space $H^{2}(\Omega)$.

We more generally define Hardy spaces $H^{p}(\Omega)$, for $1\leq p<\infty$, in the usual way:

[TABLE]

where $f$ is analytic in $\Omega$ and $\|f\|_{p}=\sup_{0<r<1}M_{p}(r,f)=\lim_{r\to 1^{-}}M_{p}(r,f)$ with:

[TABLE]

We have $\|f\|=\|f\|_{2}$. Moreover, $H^{q}$ contractively embeds into $H^{p}$ for $p<q$.

2.2 Singular numbers

We begin with a reminder of operator-theoretic facts. We recall that the approximation numbers $a_{n}(T)=a_{n}$ of an operator $T\colon H\to H$ (with $H$ a Hilbert space) are defined by:

[TABLE]

According to a 1957’s result of Allahverdiev (see [3], page 155), we have $a_{n}=s_{n}$, the $n$-th singular number of $T$. We also recall a basic result due to H. Weyl and one obvious consequence:

Theorem 2.1**.**

Let $T\colon H\to H$ be a compact operator with eigenvalues $(\lambda_{n})$ rearranged in decreasing order and singular numbers $(a_{n})$. Then:

[TABLE]

As a consequence:

[TABLE]

2.3 Spectra of projective tensor products

The following operator-theoretic result will play a basic role in the sequel. Let $E_{1},\ldots,E_{n}$ be Banach spaces and let $E=\otimes_{i=1}^{n}E_{i}$ their projective tensor product (the only tensor product we shall use). If $T_{i}\in\mathcal{L}(E_{i})$, we define as usual their projective tensor product $T=\otimes_{i=1}^{n}T_{i}\in\mathcal{L}(E)$ by its action on the atoms of $E$, namely:

[TABLE]

Denote in general $\sigma(x)$ the spectrum of $x\in\mathcal{A}$ where $\mathcal{A}$ is a unital Banach algebra. We recall ([13], chap.11, Theorem 11.23) the following result.

Lemma 2.2**.**

Let $\mathcal{A}$ be a unital Banach algebra, and $x_{1},\ldots,x_{n}$ be pairwise commuting elements of $\mathcal{A}$. Then:

[TABLE]

Here, $\prod_{i=1}^{n}\sigma(x_{i})$ is the product in the Minkowski sense, namely:

[TABLE]

As a consequence, we then have the following lemma due to Schechter, which we prove under a weakened form, sufficient here, and which is indeed already in [1] (we just add a few details because this is a central point in our estimates).

Lemma 2.3**.**

Let $F$ be a Banach space, $T_{1},\ldots,T_{n}\in\mathcal{L}(F)$ and $T=\otimes_{i=1}^{n}T_{i}$. Then $\sigma(T)\subset\prod_{i=1}^{n}\sigma(T_{i})$.

Proof.

To save notation, we assume $n=2$. Let $x_{1}=T_{1}\otimes I_{2}$ and $x_{2}=I_{1}\otimes T_{2}$ where $I_{i}$ is the identity of $E_{i}$. Clearly,

[TABLE]

where the spectrum of $x_{i}$ is in the Banach algebra $\mathcal{L}(E)$ and that of $T_{i}$ in $\mathcal{L}(E_{i})$. Lemma 2.2 now gives:

[TABLE]

hence the result. ∎

2.4 Schur maps and composition operators

We now pass to some general facts on composition operators $C_{\varphi}$, defined by $C_{\varphi}(f)=f\circ\varphi$, associated with a Schur map, namely a non-constant analytic self-map $\varphi$ of $\Omega$. We say that $\varphi$ is a symbol for $H^{2}(\Omega)$ if $C_{\varphi}$ is a bounded linear operator from $H^{2}(\Omega)$ into itself.

The differential $\varphi^{\prime}(a)$ of $\varphi$ at some point $a\in\Omega$ is a bounded linear map $\varphi^{\prime}(a)\colon\ell^{1}\to\ell^{1}$.

Definition 2.4**.**

The symbol $\varphi$ is said to be truly infinite-dimensional if the differential $\varphi^{\prime}(a)$ is an injective linear map from $\ell^{1}$ into itself for at least one point $a\in\Omega$.

In finite dimension, this amounts to saying that $\varphi(\Omega)$ has non-void interior.

We have the following general result.

Proposition 2.5**.**

Let $(\varphi_{j})_{j\geq 1}$ be a sequence of analytic self-maps of  $\mathbb{D}$ such that $\sum_{j\geq 1}|\varphi_{j}(0)|<\infty$. Then, the mapping $\varphi\colon\Omega\to\mathbb{C}^{\infty}$ defined by the formula $\varphi(z)=(\varphi_{j}(z_{j}))_{j\geq 1}$ maps $\Omega$ to itself and is a symbol for $H^{2}(\Omega)$.

Proof.

First, the Schwarz inequality:

[TABLE]

shows that $\varphi(z)\in\Omega$ when $z\in\Omega$. To see that $\varphi$ is moreover a symbol for $H^{2}(\Omega)$, we use the fact ([8]) that:

[TABLE]

Now, by the separation of variables and Fubini’s theorem, we easily get:

[TABLE]

As $\sum_{j\geq 1}|\varphi_{j}(0)|<\infty$, by hypothesis, the infinite product

[TABLE]

converges and, in view of (2.2) and (2.3), $C_{\varphi}$ is bounded. ∎

We also have the following useful fact.

Lemma 2.6**.**

The automorphisms of $\Omega$ act transitively on $\Omega$ and define bounded composition operators on $H^{2}(\Omega)$.

Proof.

Let $a=(a_{j})_{j}\in\Omega$ and let $\Psi_{a}\colon\Omega\to\mathbb{C}^{\infty}$ be defined by:

[TABLE]

where in general $\Phi_{u}\colon\mathbb{D}\to\mathbb{D}$ is defined by $\Phi_{u}(z)=(z-u)/(1-\overline{u}z)$. The Schwarz lemma gives $|\Phi_{a_{j}}(z_{j})+a_{j}|\leq 2|z_{j}|$, and shows that $\Psi_{a}$ maps $\Omega$ to itself. Clearly, $\Psi_{a}$ is an automorphism of $\Omega$ with inverse $\Psi_{-a}$ and $\Psi_{a}(a)=0$. The fact that the composition operator $C_{\Psi_{a}}$ is bounded on $H^{2}(\Omega)$ is a consequence of Proposition 2.5. ∎

3 Spectrum of compact composition operators

We begin with the following definition, following [10].

Definition 3.1**.**

Let $\varphi\colon\Omega\to\Omega$ be a truly infinite-dimensional symbol. We say that $\varphi$ is compact if $\overline{\varphi(\Omega)}$ is a compact subset of $\Omega$.

We then have the following result.

Lemma 3.2**.**

If $\varphi\colon\Omega\to\Omega$ is a compact mapping, then:

$1)$* $C_{\varphi}\colon H^{2}(\Omega)\to H^{2}(\Omega)$ is bounded and moreover compact.*

$2)$* If $a\in\Omega$ a fixed point of $\varphi$, $\varphi^{\prime}(a)\in\mathcal{L}(\ell^{1})$ is a compact operator.*

Proof.

$1)$ follows from a H. Schwarz type criterion via an Ascoli-Montel type theorem: every sequence $(f_{n})$ of $H^{2}(\Omega)$ bounded in norm contains a subsequence which converges uniformly on compact subsets of $\Omega$. Indeed, we have the following ([4], chap. 17, p. 274): if $A$ is a locally bounded set of holomorphic functions on $\Omega$, then $A$ is locally equi-Lipschitz, namely every point $a\in\Omega$ has a neighourhood $U\subset\Omega$ such that:

[TABLE]

The Ascoli-Montel theorem easily follows from this. Then, if $f_{n}\in H^{2}(\Omega)$ converges weakly to [math], it converges uniformly to [math] on compact subsets of $\Omega$; in particular on $\overline{\varphi(\Omega)}$. This means that $\|C_{\varphi}(f_{n})\|_{\infty}=\|f_{n}\circ\varphi\|_{\infty}\to 0$, implying $\|f_{n}\circ\varphi\|_{2}\to 0$ and the compactness of $C_{\varphi}$.

Actually, $C_{\varphi}$ is compact on every Hardy space $H^{p}(\Omega),\ 1\leq p\leq\infty$. This observation will be useful later on.

For $2)$, we may indeed dispense ourselves with the invariance of $a$, and force $a=0$ to be a fixed point of $\varphi$. Indeed, we can replace $\varphi$ by $\psi=\Psi_{b}\circ\varphi\circ\Psi_{a}$ where $b=\varphi(a)$ is arbitrary, and use Lemma 2.6 as well as the ideal property of compact linear operators. We set $A=\varphi^{\prime}(0)$. Expanding each coordinate $\varphi_{j}$ of $\varphi$ in a series of homogeneous polynomials, we may write (since $\varphi(0)=0$):

[TABLE]

where $c_{\alpha}=(c_{\alpha,j})_{j\geq 1}\in\mathbb{C}^{\infty}$. We clearly have (looking at the Fourier series of $\varphi(z\,{\rm e}^{i\theta})$):

[TABLE]

Since $\varphi$ is compact, this clearly implies, with $B$ the open unit ball of $\ell^{1}$, that $A(B)$ is totally bounded, proving the compactness of $A$. ∎

The following extension of results of [11], then [1] and [6], which themselves extend a theorem of G. Königs ([14], p. 93) will play an essential role for lower bounds of approximation numbers.

Theorem 3.3**.**

Let $\varphi\colon\Omega\to\Omega$ be a compact symbol. Assume there is $a\in\Omega$ such that $\varphi(a)=a$ and that $\varphi^{\prime}(a)\in\mathcal{L}(\ell^{1})$ is injective. Then, the spectrum of $C_{\varphi}\colon H^{2}(\Omega)\to H^{2}(\Omega)$ is exactly formed by the numbers $\lambda^{\alpha}$, $\alpha\in\mathbb{N}^{(\infty)}$, and $0,1$, where $(\lambda_{j})_{j\geq 1}$ denote the eigenvalues of $A:=\varphi^{\prime}(a)$ and:

[TABLE]

Proof.

This is proved in [1] for the unit ball $B_{E}$ of an arbitrary Banach space $E$ and for the space $H^{\infty}(B_{E})$, in four steps which are the following:

$1.$ If $\varphi(B_{E})$ lies strictly inside $B_{E}$ (namely if $\varphi(B_{E})\subseteq rB_{E}$ for some $r<1$), in particular when $\varphi$ is compact, $\varphi$ has a unique fixed point $a\in B_{E}$, according to a theorem of Earle and Hamilton.

$2.$ The spectrum of $C_{\varphi}$ contains the numbers $\lambda$ where $\lambda$ is an eigenvalue of $\varphi^{\prime}(a)$ or $\lambda=0,1$.

$3.$ It is then proved that the spectrum of $C_{\varphi}$ contains the numbers $\lambda^{\alpha}$ and $0,1$.

$4.$ It is finally proved that spectrum of $C_{\varphi}$ is contained in the numbers $\lambda^{\alpha}$ and $0,1$.

Here, handling with the domain $\Omega$, we see that:

$1.$ True or not for $\Omega$, the Earle-Hamilton theorem is not needed since we will force, by a change of the compact symbol $\varphi$ in another compact symbol $\psi=\Psi_{b}\circ\varphi\circ\Psi_{a}$, the point [math] to be a fixed point. Moreover $A=\psi^{\prime}(0)$ is injective if $\varphi^{\prime}(a)$ is, since $\Psi^{\prime}_{a}$ and $\Psi^{\prime}_{b}$ are invertible.

$2.$ The second step (non-surjectivity) is valid for any domain and for $H^{2}(\Omega)$, or $H^{p}(\Omega)$, in exactly the same way.

$3.$ The third step consists of proving $\{\lambda^{\alpha}\}\subseteq\sigma(C_{\varphi})$.

For that purpose, assume that $\lambda^{\alpha}=\prod_{l=1}^{m}\lambda_{l}\neq 0$ with $\lambda_{l}$ an eigenvalue of $\varphi^{\prime}(0)$ and with repetitions allowed. As we already mentioned, under the assumption of compactness of $\varphi$, $C_{\varphi}$ is compact on $H^{p}(\Omega)$ as well, for any $p\geq 1$. We take here $p=2m$. Step $2$ provides us with non-zero functions $f_{i}\in H^{p}(\Omega)$ such that $f_{i}\circ\varphi=\lambda_{i}f_{i}$, $1\leq i\leq m$, since for the compact operator $C_{\varphi}\colon H^{p}\to H^{p}$, non-surjectivity implies non-injectivity. Let $f=\prod_{1\leq i\leq m}f_{i}$. Then, using the integral representation of the norm and the Hölder inequality, we see that $f\in H^{2}(\Omega)$, $f\neq 0$ and $f\circ\varphi=\lambda^{\alpha}f$, proving our claim.

$4.$ The fourth step is valid as well, with a slight simplification: we have to show that, if $\mu\neq 1$ is not of the form $\lambda^{\alpha}$, then $C_{\varphi}-\mu I$ is injective. Let $f\in H^{2}(\Omega)$ satisfying $f\circ\varphi=\mu f$ and let:

[TABLE]

be the Taylor expansion of $f$ about $z=0$ (observe that $\Omega$ is a Reinhardt domain). As usual, $d^{m}f(0)=:L_{m}$ is an $m$-linear symmetric form on $F=\ell^{1}$ and the notation $L_{m}(z^{m})$ means $L_{m}(z,z,\ldots,z)$.

Observe that $L_{m}$ can be isometrically identified with an element (denoted $\overline{L_{m}}$) of $\mathcal{L}(F^{\otimes n})$ defined by the formula:

[TABLE]

We will prove by induction that $L_{n}=0$ for each $n$. For this, we can avoid the appeal to transposes of [1] as follows: if the result holds for $L_{m}$ with $m<n$, one gets (comparing the terms in $z^{n}$ in both members of $f\circ\varphi=\mu f$):

[TABLE]

That is $A(B-\mu I)=0$ where $I$ is the identity map of $F^{\otimes n}$. Now, $B-\mu I$ in invertible in $\mathcal{L}(F)$ by Lemma 3.3, so that $A=A(B-\mu I)(B-\mu I)^{-1}=0$.

The proof is complete. ∎

The following theorem summarizes and exploits the preceding theorem. Possibly, some restrictions can be removed, and we could only assume the compactness of $C_{\varphi}$, not of $\varphi$ itself. After all, in dimension one, there are symbols $\varphi$ with $\|\varphi\|_{\infty}=1$ for which $C_{\varphi}\colon H^{2}\to H^{2}$ is compact.

Theorem 3.4**.**

Let $\varphi\colon\Omega\to\Omega$ be a truly infinite-dimensional compact mapping of $\Omega$. Then:

$1)$* $C_{\varphi}:H^{2}(\Omega)\to H^{2}(\Omega)$ is bounded and even compact.*

$2)$* $A=\varphi^{\prime}(0)$ is compact.*

$3)$* No $\delta>0$ can exist such that $a_{n}(C_{\varphi})\leq C\,{\rm e}^{-c\,n^{\delta}}$ for all $n\geq 1$. More precisely, the numbers $a_{n}$ satisfy:*

[TABLE]

Proof.

The proof is based on the previous Theorem 3.3. Without loss of generality, we can assume that $\varphi(0)=0$ and $\varphi^{\prime}(0)$ is injective, by using a point $a$ at which $\varphi^{\prime}(a)$ is injective, and then the fact that automorphisms of $\Omega$ act transitively on $\Omega$, act boundedly on $H^{2}(\Omega)$, and the ideal property of approximation numbers. More precisely, we pass to $\Psi=\Psi_{b}\circ\varphi\circ\Psi_{a}$ with $b=\varphi(a)$ and get:

[TABLE]

injective, since $\Psi^{\prime}_{b}(b)$ and $\Psi^{\prime}_{a}(0)$ are, and $\Psi_{a}$ and $\Psi_{b}$ are automorphisms of $\Omega$.

We now have the following simple but crucial lemma.

Lemma 3.5**.**

Whatever the choice of the numbers $\lambda_{j}$ with $0<|\lambda_{j}|<1$, denoting by $(\delta_{n})_{n\geq 1}$ the non-increasing rearrangement of the numbers $\lambda^{\alpha}$, one has:

[TABLE]

Proof of the Lemma.

For any positive integer $p$, we set:

[TABLE]

and we use that:

[TABLE]

where $\|\,.\,\|$ stands for the euclidean norm in $\mathbb{R}^{q}$. We then get:

[TABLE]

because:

[TABLE]

This proves the lemma. ∎

This can be transferred to the approximation numbers $a_{n}=a_{n}(C_{\varphi})$ to end the proof of Theorem 3.4. Indeed, we know from Lemma 3.5 that the non-increasing rearrangement $(\delta_{n})$ of the eigenvalues $\lambda^{\alpha}$ of $C_{\varphi}$ satisfies

[TABLE]

Since a divergent series of non-negative and non-increasing numbers $u_{n}$ satisfies $\sum u_{2n}=\infty$, we further see that:

[TABLE]

Moreover, by Theorem 2.1 we have:

[TABLE]

Since $1/(\log 1/a_{1}a_{n})\sim 1/(\log 1/a_{n})$, Lemma 3.5 then gives the result. This clearly prevents an inequality of the form $a_{n}\leq C\,{\rm e}^{-c\,n^{\delta}}$ for some positive numbers $c,C,\delta$ and all $n\geq 1$. Indeed, this would imply:

[TABLE]

contradicting (3.3). ∎

Remarks. Let us briefly comment on the assumptions in Theorem 3.4.

$1)$ We do not need the Earle-Hamilton theorem under our assumptions. The Schauder-Tychonoff theorem gives the existence (if not the uniqueness) of a fixed point for $\varphi$ in $\Omega$ (bounded and convex).

$2)$ The Earle-Hamilton theorem is in some sense more general (for analytic maps) since it remains valid when $\overline{\varphi(\Omega)}$ is only assumed to lie strictly inside $\Omega$, i.e. when $\varphi(\Omega)\subseteq r\Omega$ for some $r<1$. But this assumption does not ensure the compactness of $C_{\varphi}$ as indicated by the simple example $\varphi(z)=rz$, $0<r<1$. The coordinate functions $z\mapsto z_{n}$ converge weakly to [math], while $\|C_{\varphi}(z_{n})\|_{H^{2}(\Omega)}=r$.

$3)$ The mere assumption that $\overline{\varphi(\Omega)}$ is compact is not sufficient either. Juste take:

[TABLE]

Since the composition operator $C_{\varphi_{1}}$ associated with $\varphi_{1}(z)=\frac{1+z}{2}$ is notoriously non-compact on $H^{2}(\mathbb{D})$, neither is $C_{\varphi}$ on $H^{2}(\Omega)$. Yet, $\overline{\varphi(\Omega)}$ is obviously compact in $\ell^{1}$.

4 Possible upper bounds

Recall that $\Omega=\mathbb{D}^{\infty}\cap\ell^{1}$.

4.1 A general example

Theorem 4.1**.**

Let $\varphi((z_{j})_{j})=(\lambda_{j}z_{j})_{j}$ with $|\lambda_{j}|<1$ for all $j$, so that $\varphi(\Omega)\subseteq\Omega$ and $\varphi^{\prime}(0)$ is the diagonal operator with eigenvalues $\lambda_{j}$, $j\geq 1$, on the canonical basis of $\ell^{1}$. Let $p>0$. Then:

[TABLE]

*In particular, there exist truly infinite-dimensional symbols on $\Omega$ such that the composition operator $C_{\varphi}\colon H^{2}(\Omega)\to H^{2}(\Omega)$ is in all Schatten classes $S_{p}$, $p>0$. *

Proof.

Since $C_{\varphi}$ is diagonal on the orthonormal basis $(z^{\alpha})_{\alpha}$ of the Hilbert space $H^{2}(\Omega)$, with $C_{\varphi}(z^{\alpha})=\varphi^{\alpha}$, its approximation numbers are the non-increasing rearrangement of the moduli of eigenvalues $\lambda^{\alpha}$, so that an Euler product-type computation gives:

[TABLE]

To obtain $C_{\varphi}\in\bigcap_{p>0}S_{p}$, just take $\lambda_{n}={\rm e}^{-n}$. This ends the proof. ∎

4.2 A sharper upper bound

By making a more quantitative study, we can prove the following result.

Theorem 4.2**.**

For any $0<\delta<1$, there exists a compact composition operator on $H^{2}(\Omega)$, with a truly infinite-dimensional symbol, such that, for some positive constants $c,C,b$, we have:

[TABLE]

Proof.

Take the same operator $C_{\varphi}$ as in Theorem 4.1, with $\lambda_{n}={\rm e}^{-A_{n}}$ where the positive numbers $A_{n}$ have to be adjusted. Its approximation numbers $a_{N}$ are then the non-increasing rearrangement of the sequence of numbers $(\varepsilon_{n})_{n}:=(\lambda^{\alpha})_{\alpha}$. This suggests using a generating function argument, namely considering $\sum\varepsilon_{n}x^{n}$, but the rearrangement perturbs the picture. Accordingly, we follow a sligthly different route. Fix an integer $N\geq 1$ and a real number $r>0$. Observe that, following the proof of Theorem 4.1:

[TABLE]

First, consider the simple example $A_{n}=n$. We get:

[TABLE]

where $\eta$ is the Dedekind eta function (see [5]) given by:

[TABLE]

where $p(n)$ is the number of partitions of the integer $n$. It is well-known ([5], Ch. 7, p. 169) that $\eta\,({\rm e}^{-r})\leq{\rm e}^{D/r}$ with $D=\pi^{2}/6$, so that:

[TABLE]

Optimizing with $r=2D/\log N$, we get:

[TABLE]

with $c=1/4D$. This is more precise than Theorem 4.1.

We now show that if $A_{n}$ increases faster, we can achieve the decay of Theorem 4.2. As before, we get in general:

[TABLE]

where

[TABLE]

We have:

[TABLE]

Now, take $A_{n}={\rm e}^{n^{\alpha}}$ where $\alpha>0$ is to be chosen. We have:

[TABLE]

Standard estimates now give, for $r<1$:

[TABLE]

so that:

[TABLE]

Going back to (4.1), we get, for some constant $C>0$, and for $x=1/r>1$:

[TABLE]

Adjusting $x=x_{N}>1$ so as to have $(\log x)^{\frac{1}{\alpha}+1}=\log N-1$, that is:

[TABLE]

we get $a_{N}\leq C\,{\rm e}^{-c\,x_{N}}$, which is the claimed result with $\delta=\alpha/(\alpha+1)$.

This $\delta$ can be taken arbitrarily in $(0,1)$ by choosing $\alpha$ suitable, and we are done. ∎

Remark. Of course, $\delta=1$ is forbidden, because this would give $a_{n}\leq C\,{\rm e}^{-c\,n^{b}}$, implying:

[TABLE]

for large $p$, and contradicting Theorem 3.4.