Dixmier Trace for Toeplitz Operators on Symmetric Domains

TL;DR

This paper introduces a new approach to analyzing Toeplitz operators on symmetric domains by defining a Hilbert quotient module and deriving explicit formulas for their Dixmier traces, linking operator theory with boundary geometry.

Contribution

It defines a Hilbert quotient module for Toeplitz operators on symmetric domains and provides an explicit formula for their Dixmier trace based on boundary geometry.

Findings

Hilbert quotient module belongs to Macaev class ${ ext{L}}^{n, ext{infty}}$

Explicit Dixmier trace formula for Toeplitz commutators

Connection between boundary geometry and operator traces

Abstract

For Toeplitz operators on bounded symmetric domains of arbitrary rank, we define a Hilbert quotient module corresponding to partitions of length $1$ and prove that it belongs to the Macaev class ${\mathcal{L}}^{n,\infty}$. We next obtain an explicit formula for the Dixmier trace of Toeplitz commutators in terms of the underlying boundary geometry.