A note on semi-Fredholm Hilbert modules

TL;DR

This paper investigates the spectral properties of Toeplitz-like operators on Hilbert spaces of vector-valued holomorphic functions, providing necessary conditions for their Fredholmness and establishing sufficiency in the unit disk case.

Contribution

It introduces necessary conditions for k-tuples of Toeplitz-like operators to be Fredholm and proves their sufficiency specifically for the unit disk.

Findings

Necessary conditions for Fredholmness of operator tuples

Sufficiency of these conditions in the unit disk case

Advances understanding of spectral theory for Toeplitz-like operators

Abstract

A classical problem in operator theory has been to determine the spectrum of Toeplitz-like operators on Hilbert spaces of vector-valued holomorphic functions on the open unit ball in C^m. In this note we obtain necessary conditions for k-tuples of such operators to be Fredholm in the sense of Taylor and show they are sufficient in the case of the unit disk.