The group of isometries of a Banach space and duality

TL;DR

This paper constructs examples of Banach spaces illustrating contrasting properties of their isometry groups and duals, revealing complex behaviors in symmetry groups and operator classes.

Contribution

It provides the first known example of a Banach space with a non-regular isometry group but a dual with infinitely many continuous one-parameter groups.

Findings

The constructed Banach space's isometry group lacks uniformly continuous one-parameter semigroups.

Its dual's isometry group contains infinitely many such semigroups.

Additional examples relate to numerical index, hermitian, and dissipative operators.

Abstract

We construct an example of a real Banach space whose group of surjective isometries has no uniformly continuous one-parameter semigroups, but the group of surjective isometries of its dual contains infinitely many of them. Other examples concerning numerical index, hermitian operators and dissipative operators are also shown.