On the local spectrum and the adjoint

James Kenneth Finch

Vol. 94 (1981), No. 2, 297–302

DOI: 10.2140/pjm.1981.94.297

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 x^∗x = 0.

Mathematical Subject Classification 2000

Primary: 47A10

Milestones

Received: 12 October 1979

Published: 1 June 1981