About radial Toeplitz operators on Segal-Bargmann and $l^2$ spaces

TL;DR

This paper explores the relationship between radial Toeplitz operators on the Segal-Bargmann space and diagonal operators on l^2, establishing conditions for their equivalence and analyzing inverse and composition problems.

Contribution

It introduces conditions for equivalence between Toeplitz and diagonal operators in the radial case and examines the inverse mapping from diagonal to Toeplitz operators.

Findings

Established isometric mapping between Segal-Bargmann space and l^2 for radial symbols.

Derived conditions for Toeplitz operators to be equivalent to diagonal operators on l^2.

Analyzed the inverse problem of representing diagonal operators as Toeplitz operators.

Abstract

We discuss Toeplitz operators on the Segal-Bargmann space as functional realizations of anti-Wick operators on the Fock space. In the special case of radial symbols we exploit the isometric mapping between the Segal-Bargmann space and $l^2$ complex sequences in order to establish conditions such that an equivalence between Toeplitz operators and diagonal operators on $l^2$ holds. We also analyze the inverse problem of mapping diagonal operators on $l^2$ into Toeplitz form. The composition problem of Toeplitz operators with radial symbols is reviewed as an application. Our notation and basic examples make contact with Quantum Mechanics literature.