A Toeplitz-like operator with rational symbol having poles on the unit circle I: Fredholm properties

TL;DR

This paper investigates a class of unbounded Toeplitz-like operators with rational symbols having poles on the unit circle, establishing conditions for Fredholmness and providing an index formula.

Contribution

It introduces a definition for unbounded Toeplitz-like operators with poles on the unit circle and characterizes their Fredholm properties and index.

Findings

Operator is Fredholm iff the symbol has no zeros on the unit circle

A formula for the operator's index is provided

Matrix representation of the operator is discussed

Abstract

In this paper a definition is given for an unbounded Toeplitz-like operator with rational symbol which has poles on the unit circle. It is shown that the operator is Fredholm if and only if the symbol has no zeroes on the unit circle, and a formula for the index is given as well. Finally, a matrix representation of the operator is discussed.