Any power of an isometry is again an isometry. What about the converse of this fact? More precisely, is it true that a given isometry is a power of some isometry (with respect to the same norm)? In other words, for a given positive integer k, does there exist an isometric kth root of a given isometry? The aim of this talk is to answer this question in the setting of various normed spaces. It will be shown that the answer depends on a given norm and a given power k, it differs in real and in complex spaces, and it also differs in finite and in infinite dimensions. In particular, examples of isometries with and without isometric roots will be presented.
The talk is based on joint work with Bojan Kuzma from the University of Primorska, Koper, Slovenia.
The work of Dijana Ilišević has been fully supported by the Croatian Science Foundation under the project