Journal of Operator Theory

Volume 65, Issue 2, Spring 2011 pp. 255-279.

AC$(\sigma)$ operators

Authors: Brenden Ashton (1) and Ian Doust (2)
Author institution: (1) Silverbrook Research, 3 Montague St, Balmain 2041, Australia
(2) School of Mathematics and Statistics, University of New South Wales, UNSW Sydney 2052, Australia

Summary: 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 non-empty 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.

Keywords: functions of bounded variation, absolutely continuous functions, functional calculus, $AC$-operators, well-bounded operators

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