Toeplitz operators in the Herglotz space

TL;DR

This paper develops a new framework for Toeplitz operators in the Herglotz space of solutions to the Helmholtz equation, using sesquilinear forms and reproducing kernels, and explores their key properties.

Contribution

It introduces a novel approach to defining Toeplitz operators in the Herglotz space, overcoming the lack of a Bergman projection, and analyzes their fundamental properties.

Findings

Unique determination of symbols from operators

Conditions for boundedness and compactness

Spectral and algebraic properties of the operators

Abstract

We define and study Toeplitz operators in the space of Herglotz solutions of the Helmholtz equation in $R^d$. As the most traditional definition of Toeplitz operators via Bergman-type projection is not available here, we use an approach based upon the reproducing kernel nature of the Herglotz space and sesquilinear forms, which results in a meaningful theory. For two important patterns of sesquilinear forms we discuss a number of properties, including the uniqueness of determining the symbols from the operator, the finite rank property, the conditions for boundedness and compactness, spectral properties, certain algebraic relations.