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Toeplitz operators on non-reflexive Fock spaces

Robert Fulsche

Abstract

We generalize several results on Toeplitz operators over reflexive, standard weighted Fock spaces $F_t^p$ to the non-reflexive cases $p = 1, \infty$. Among these results are the characterization of compactness and the Fredholm property of such operators, a well-known representation of the Toeplitz algebra, a characterization of the essential centre of the Toeplitz algebra. Further, we improve several results related to correspondence theory, e.g. we improve previous results on the correspondence of algebras and we give a correspondence theoretic version of the well-known Berger-Coburn estimates.

Toeplitz operators on non-reflexive Fock spaces

Abstract

We generalize several results on Toeplitz operators over reflexive, standard weighted Fock spaces to the non-reflexive cases . Among these results are the characterization of compactness and the Fredholm property of such operators, a well-known representation of the Toeplitz algebra, a characterization of the essential centre of the Toeplitz algebra. Further, we improve several results related to correspondence theory, e.g. we improve previous results on the correspondence of algebras and we give a correspondence theoretic version of the well-known Berger-Coburn estimates.
Paper Structure (12 sections, 49 theorems, 145 equations)

This paper contains 12 sections, 49 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Correspondence Theorem
  4. Applications of the Correspondence Theorem
  5. Correspondence of algebras
  6. On the Fredholm property
  7. Some auxiliary facts
  8. The algebra of limit operators
  9. Reconstructing operators from limit operators
  10. Fredholmness through invertibility of limit operators
  11. Discussion
  12. Acknowledgement

Key Result

Theorem 1

Let $A \in \mathcal{L}(F_t^p)$, where $1 < p < \infty$. Then,

Theorems & Definitions (86)

  • Theorem : Bauer_Isralowitz2012
  • Theorem : Xia
  • Theorem : Berger_Coburn1987
  • Theorem : Fulsche_Hagger
  • Theorem
  • Lemma 2.1
  • Example
  • Theorem 3.1
  • Theorem 3.2: Berger_Coburn1994Bauer_Fulsche2020
  • Corollary 3.3
  • ...and 76 more