A Class of Toeplitz Operators in Several Complex Variables

TL;DR

This paper introduces a singular Cauchy type integral to analyze Toeplitz algebras associated with Dirac operators near smooth boundaries in complex n-space, advancing index theory.

Contribution

It develops a principal symbol computation for a new singular integral, enabling progress in the index theory of Toeplitz algebras in several complex variables.

Findings

Computed the principal symbol of the singular Cauchy integral.

Initiated the index theory for Toeplitz algebras in complex variables.

Laid groundwork for future analysis of Dirac operators near boundaries.

Abstract

In order to study the Toeplitz algebras related to a Dirac operators in a neighborhood of a closed bounded domain D with smooth boundary in C^n we introduce a singular Cauchy type integral. We compute its principal symbol, thus initiating the index theory.