# Journal of Operator Theory

Volume 86, Issue 2, Fall 2021 pp. 255-273.

On isometries with finite spectrum**Authors**: Fernanda Botelho (1), Dijana Ilisevic (2)

**Author institution:**(1) Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, U.S.A.

(2) Department of Mathematics, Faculty of Science, University of Zagreb, Zagreb, 10000, Croatia

**Summary:**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 $\mathcal{X}$ with the following property: if $T : \mathcal{X} \rightarrow \mathcal{X}$ is a linear isometry with two-point spectrum $\{1, \lambda \} $ then $\lambda = -1$ or the eigenprojections of $T$ are Hermitian.

**DOI:**http://dx.doi.org/10.7900/jot.2020apr11.2270

**Keywords:**linear isometry, periodic linear isometry, finite spectrum, eigenprojection, inverse eigenvalue problem for isometries

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