Toeplitz operators defined by sesquilinear forms: Fock space case

TL;DR

This paper introduces a unified sesquilinear form framework to extend the class of symbols defining Toeplitz operators in Fock space, allowing for highly singular symbols beyond traditional restrictions.

Contribution

It develops a broad, unified approach for defining Toeplitz operators via sesquilinear forms, extending the class of admissible symbols in Fock space.

Findings

Extended the class of symbols for Toeplitz operators in Fock space.

Unified framework covers previous cases and new singular symbols.

Established boundedness criteria for the extended class.

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.