An Index Theorem for Toeplitz Operators on the Quarter-Plane

TL;DR

This paper establishes an index theorem for Toeplitz operators on the quarter-plane by leveraging generalized Toeplitz operator theory and constructing an indicial triple on tensor product C*-algebras, extending existing frameworks.

Contribution

It introduces a new index theorem for quarter-plane Toeplitz operators using indicial triples and extends the theory to broader classes of such operators.

Findings

Proved an index theorem for quarter-plane Toeplitz operators.

Constructed an indicial triple on tensor product C*-algebras.

Extended the theory to certain classes of Toeplitz operators.

Abstract

We prove an index theorem for Toeplitz operators on the quarter-plane using the index theory for generalized Toeplitz operators introduced by G. J. Murphy. To prove this index theorem we construct an indicial triple on the tensor product of two C*-algebras provided with indicial triples with general conditions. We show that our results can be extended to some extensions of the theory of Toeplitz operators on the quarter-plane.