Abstract. Let $\gamma $ be a unimodular complex number, and let $k$ be an integer. Then $\gamma A^k$ is an isometry for any isometry $A$ of a complex Banach space. It is shown that if $f$ is an analytic function on the unit circle sending an isometry to an isometry for any norm, then $f$ has the form $z \mapsto \gamma z^k$ for some unimodular $\gamma $ and integer $k$. The same conclusion on $f$ can be deduced if $f$ is merely continuous and preserves the isometries of some special classes of norms on a fixed finite-dimensional complex Banach space. The result is extended to real Banach spaces $X$ with $\dim {X} \geq 4$, and it is shown that one cannot get the same conclusion on $f$ if $\dim {X}<4$. Further extensions of these results are also considered.

DOI: 10.14232/actasm-017-056-6

AMS Subject Classification (1991): 47B49, 15A60, 15A86, 46B04

Keyword(s): Banach space, isometry, continuous function, generalized permutation

Received September 10, 2017 and in final form February 5, 2018. (Registered under 56/2017.)