ON LOCAL SPECTRAL PROPERTIES OF GENERALIZED SCALAR OPERATORS

Abstract

In this paper, we prove that if T ∈ L(X) is a generalized scalar operator then Ker T p is the quasi-nilpotent part of T for some positive integer p ∈ N. Moreover, we prove that a generalized scalar operator with finite spectrum is algebraic. In particular, a quasi-nilpotent generalized scalar operator is nilpotent.

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