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Analysis of Toeplitz Operators with $BMO^1_α$ operator-valued symbols on $\ell^2-$Valued Bergman Spaces

David Békollè, Hugues Olivier Défo, Edgar L. Tchoundja

TL;DR

This work analyzes Toeplitz operators on the $\ell^2$-valued Bergman space $A^2_\alpha({\mathbb B}_n, \ell^2)$ with operator-valued symbols in $BMO^1_\alpha({\mathbb B}_n, \mathcal{L}(\ell^2))$, developing a finite-dimensional reduction method and constructing explicit non-compact examples. It establishes sufficient conditions for compactness via the Toeplitz algebra $\mathcal{T}_{L^\infty_{fin}}$ and through sufficiently localized operators, and connects compactness to Berezin-transform vanishing and boundary behavior of the symbols and their adjoints. The paper also proves finite-dimensional truncation results and shows that certain extensions from the finite-dimensional case to infinite dimension fail without extra hypotheses, while offering a concrete framework and open questions for further extending Xia’s localization results to the infinite-dimensional setting. Together, these results provide a robust operator-theoretic toolkit for vector-valued Bergman spaces, enabling precise compactness criteria for $T_b$ and clarifying when finite-dimensional approximations capture infinite-dimensional behavior.

Abstract

As a class of compact operators on the $\ell^2-$valued Bergman space $A^2_α(\mathbb B_n, \ell^2)$ on the unit ball $\mathbb B_n,$ we study Toeplitz operators with $BMO^1_α(\mathbb B_n, \mathcal L(\ell^2))$ operator-valued symbols. First, we describe a method of restriction to a finite dimension which allows us to apply earlier results of Rahm and Wick; then we exhibit an explicit example of a compact Toeplitz operator on $A^2_α(\mathbb B_n, \ell^2).$ Secondly, we apply two sufficient conditions for compactness established by Rahm in infinite dimension. The first condition is in terms of the Toeplitz algebra $\mathcal T_{L^\infty_{fin}},$ the second one is in terms of sufficiently localized operators and is implied by the first condition. To get the second condition, we additionally assume that the symbol and its adjoint belong to $BMO^1_α(\mathbb B_n, \mathcal L(\ell^2, \ell^1)).$ Finally, inspired by Xia and Sadeghi-Zorboska, we ask the question of the validity of the reverse implication.

Analysis of Toeplitz Operators with $BMO^1_α$ operator-valued symbols on $\ell^2-$Valued Bergman Spaces

TL;DR

This work analyzes Toeplitz operators on the -valued Bergman space with operator-valued symbols in , developing a finite-dimensional reduction method and constructing explicit non-compact examples. It establishes sufficient conditions for compactness via the Toeplitz algebra and through sufficiently localized operators, and connects compactness to Berezin-transform vanishing and boundary behavior of the symbols and their adjoints. The paper also proves finite-dimensional truncation results and shows that certain extensions from the finite-dimensional case to infinite dimension fail without extra hypotheses, while offering a concrete framework and open questions for further extending Xia’s localization results to the infinite-dimensional setting. Together, these results provide a robust operator-theoretic toolkit for vector-valued Bergman spaces, enabling precise compactness criteria for and clarifying when finite-dimensional approximations capture infinite-dimensional behavior.

Abstract

As a class of compact operators on the valued Bergman space on the unit ball we study Toeplitz operators with operator-valued symbols. First, we describe a method of restriction to a finite dimension which allows us to apply earlier results of Rahm and Wick; then we exhibit an explicit example of a compact Toeplitz operator on Secondly, we apply two sufficient conditions for compactness established by Rahm in infinite dimension. The first condition is in terms of the Toeplitz algebra the second one is in terms of sufficiently localized operators and is implied by the first condition. To get the second condition, we additionally assume that the symbol and its adjoint belong to Finally, inspired by Xia and Sadeghi-Zorboska, we ask the question of the validity of the reverse implication.
Paper Structure (27 sections, 47 theorems, 124 equations)

This paper contains 27 sections, 47 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. The Space $BMO^1_{\alpha} ({\mathbb B}_n, \mathcal{L}(\ell^2))$
  3. Preliminary results
  4. A class of examples of non compact Toeplitz operators with $BMO^1_\alpha (\mathbb B_n, \mathcal{L}(\ell^2))$ operator-valued symbols on $A^2_\alpha (\mathbb B_n, \ell^2)$
  5. Three preliminary lemmas
  6. Toeplitz operators with symbols in $BMO^1_{\alpha} ({\mathbb B}_n, \mathcal{L}(\ell^2)).$
  7. A preliminary remark
  8. A sufficient condition on the symbol $b$ for the estimate $\lim \limits_{d\rightarrow \infty} \left \Vert T_b M_{I_{(d)}}\right \Vert_{\mathcal{L}(A^2_\alpha (\mathbb B_n, \mathcal{L}(\ell^2)))}=0$
  9. A sufficient condition on the symbol $b$ for the estimate $\lim \limits_{d\rightarrow \infty} \left \Vert M_{I_{(d)}}T_b \right \Vert_{\mathcal{L}(A^2_\alpha (\mathbb B_n, \ell^2))}=0$
  10. A method of reduction to the finite dimension
  11. A class of examples of compact Toeplitz operators on $A^2_\alpha (\mathbb B_n, \ell^2)$ with symbols in $BMO^1_{\alpha} ({\mathbb B}_n, \mathcal{L}(\ell^2))$
  12. Sufficient conditions for the compactness of the Toeplitz operator $T_b$ with a symbol $b$ in $BMO^1_{\alpha} ({\mathbb B}_n, \mathcal{L}(\mathbb C^d))$
  13. Proof of Theorem 1.10
  14. Proof of Theorem \ref{['thm111']}
  15. Proof of Theorem 1.13
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $E$ and $F$ be two complex Banach spaces and let $T: E\rightarrow F$ be a continuous linear operator. Let $\mu$ be a positive $\sigma-$finite Borel measure on ${\mathbb B}_n.$ If $f: {\mathbb B}_n \rightarrow E$ is $\mu-$Bochner integrable, then $T(f)$ is also $\mu-$Bochner integrable and we hav

Theorems & Definitions (77)

  • Theorem 1.1: Hille's Theorem
  • Proposition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Definition 1.7
  • Definition 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 67 more