Non--Vanishing functions and Toeplitz Operators on Tube--Type Domains

TL;DR

This paper establishes an index theorem for Toeplitz operators on tube-type domains, extending to matrix symbols, and characterizes non-vanishing boundary functions as products of norms and exponentials.

Contribution

It introduces a new index theorem for Toeplitz operators on tube-type domains and characterizes boundary functions in terms of generic norms and exponentials.

Findings

Proved an index theorem for Toeplitz operators on irreducible tube-type domains.

Extended results to Toeplitz operators with matrix symbols.

Characterized non-vanishing boundary functions as products of powers of norms and exponentials.

Abstract

We prove an index theorem for Toeplitz operators on irreducible tube--type domains and we extend our results to Toeplitz operators with matrix symbols. In order to prove our index theorem, we proved a result asserting that a non--vanishing function on the Shilov boundary of a tube--type bounded symmetric domain, not necessarily irreducible, is equal to a unimodular function defined as the product of powers of generic norms times an exponential function.