Naslov
Preservers of isometries

Autori
Ilišević, Dijana ; Kuzma, Bojan ; Li, Chi-Kwong ; Poon, Edward

Izvornik
Acta scientiarum mathematicarum (0001-6969) 84 (2018), 1-2; 3-17

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
Banach space ; isometry ; continuous function ; generalized permutation

Sažetak
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.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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