Vol. 94, No. 2, 1981

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On the local spectrum and the adjoint

James Kenneth Finch

Vol. 94 (1981), No. 2, 297–302
Abstract

Let X be a Banach space, X the dual space, and suppose that T is a closed linear operator on X. Assume that the domain of T is dense in X, so that the adjoint operator T is a closed linear operator on X. The local spectrum σ(T,x) is defined below. In this paper we investigate some of the relations between σ(T,x) and σ(T,x). In particular we show that if σ(T,x) and σ(T,x) are both empty, then xx = 0.

Mathematical Subject Classification 2000
Primary: 47A10
Milestones
Received: 12 October 1979
Published: 1 June 1981
Authors
James Kenneth Finch