Journal of Operator Theory
Volume 65, Issue 2, Spring 2011 pp. 255-279.
AC(σ) operatorsAuthors: 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 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.
Keywords: functions of bounded variation, absolutely continuous functions, functional calculus, AC-operators, well-bounded operators
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