$AC(σ)$ operators

Details $AC(σ)$ operators $AC(σ)$ operators

2008-07-07

arxiv.org/abs/0807.1045v2

J. Operator Theory, 65 (2011), 255-279

Mathematics

Functional Analysis

This version corrects a number of typographic errors, as well as filling in some missing details in some of the proofs

Scientific paper

In this paper we present a new extension of the theory of well-bounded operators to cover operators with complex spectrum. In previous work a new concept of the class of absolutely continuous functions on a nonempty compact subset $\sigma$ of the plane, denoted $AC(\sigma)$, was introduced. An $\AC(\sigma)$ operator is one which admits a functional calculus for this algebra of functions. The class of $AC(\sigma)$ operators includes all of the well-bounded operators and trigonometrically well-bounded operators, as well as all scalar-type spectral operators, but is strictly smaller than Berkson and Gillespie's class of $AC$ operators. This paper develops the spectral properties of $AC(\sigma)$ operators and surveys some of the problems which remain in extending results from the theory of well-bounded operators.

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