Compact $AC(\sigma)$ operators

TL;DR

This paper explores the structure of compact $AC(\sigma)$ operators, providing a representation similar to compact normal operators and conditions for constructing many such operators, advancing understanding of their decomposition.

Contribution

It introduces a representation for compact $AC(\sigma)$ operators akin to that of compact normal operators and offers conditions for constructing numerous such operators.

Findings

Compact $AC(\sigma)$ operators have a similar representation to compact normal operators.

Conditions are established for constructing a large class of such operators.

The paper addresses questions about decomposing compact $AC(\sigma)$ operators into real and imaginary parts.

Abstract

All compact $AC(\sigma)$ operators have a representation analogous to that for compact normal operators. As a partial converse we obtain conditions which allow one to construct a large number of such operators. Using the results in the paper, we answer a number of questions about the decomposition of a compact $AC(\sigma)$ into real and imaginary parts.