On isometries with finite spectrum

Abstract

In this paper we investigate inverse eigenvalue problems for finite spectrum linear isometries on complex Banach spaces. We establish necessary conditions on a finite set of modulus one complex numbers to be the spectrum of a linear isometry. In particular, we study periodic linear isometries on the large class of Banach spaces X with the following property: if T:X→X is a linear isometry with two-point spectrum {1,λ} then λ=−1 or the eigenprojections of T are Hermitian.