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Toeplitz operators defined by sesquilinear forms: Fock space case

Grigori Rozenblum, Nikolai Vasilevski

Abstract

The classical theory of Toeplitz operators in spaces of analytic functions deals usually with symbols that are bounded measurable functions on the domain in question. A further extension of the theory was made for symbols being unbounded functions, measures, and compactly supported distributions, all of them subject to some restrictions. In the context of a reproducing kernel Hilbert space we propose a certain framework for a `maximally possible' extension of the notion of Toeplitz operators for a `maximally wide' class of `highly singular' symbols. Using the language of sesquilinear forms we describe a certain common pattern for a variety of analytically defined forms which, besides covering all previously considered cases, permits us to introduce a further substantial extension of a class of admissible symbols that generate bounded Toeplitz operators. Although our approach is unified for all reproducing kernel Hilbert spaces, for concrete operator consideration in this paper we restrict ourselves to Toeplitz operators acting on the standard Fock (or Segal-Bargmann) space.

Toeplitz operators defined by sesquilinear forms: Fock space case

Abstract

The classical theory of Toeplitz operators in spaces of analytic functions deals usually with symbols that are bounded measurable functions on the domain in question. A further extension of the theory was made for symbols being unbounded functions, measures, and compactly supported distributions, all of them subject to some restrictions. In the context of a reproducing kernel Hilbert space we propose a certain framework for a `maximally possible' extension of the notion of Toeplitz operators for a `maximally wide' class of `highly singular' symbols. Using the language of sesquilinear forms we describe a certain common pattern for a variety of analytically defined forms which, besides covering all previously considered cases, permits us to introduce a further substantial extension of a class of admissible symbols that generate bounded Toeplitz operators. Although our approach is unified for all reproducing kernel Hilbert spaces, for concrete operator consideration in this paper we restrict ourselves to Toeplitz operators acting on the standard Fock (or Segal-Bargmann) space.

Paper Structure

This paper contains 14 sections, 19 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. General operator theory approach to Toeplitz operators
  3. Non-Toeplitz operators in $\mathcal{F}^2(\mathbb{C})$
  4. Classes of unbounded symbols
  5. Radial operators
  6. Finite rank operators
  7. General sesquilinear forms
  8. Estimates for derivatives of functions in the Fock space
  9. Fock-Carleson measures for derivatives; norm estimates
  10. Sharper estimates
  11. Sesquilinear forms of infinite differential order and corresponding Toeplitz operators
  12. Forms of a finite order
  13. Symbols of weak almost-finite type
  14. Symbols of norm almost-finite type

Key Result

Lemma 2.5

For each $p,\,q \in \mathbb{Z}_+$ the rank one operator $P_{p,q}$ is a Toeplitz operator with a certain distributional symbol having the one-point support $\{0\}$.

Theorems & Definitions (55)

  • Remark 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Lemma 2.5
  • Example 3.1
  • Example 3.2
  • Proposition 3.3
  • Example 3.4
  • Example 3.5
  • ...and 45 more