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$AC(σ)$ operators

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 $σ$ of the plane, denoted $AC(σ)$, was introduced. An $\AC(σ)$ operator is one which admits a functional calculus for this algebra of functions. The class of $AC(σ)$ 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 $AC(σ)$ operators and surveys some of the problems which remain in extending results from the theory of well-bounded operators.