Category theorems for stable semigroups

TL;DR

This paper extends classical category theorems from measure-preserving transformations to the setting of unitary and isometric C_0-semigroups on Hilbert spaces, establishing topological size results.

Contribution

It proves that weakly stable semigroups form a first category set, while almost weakly stable semigroups form a residual set in this context.

Findings

Weakly stable unitary groups are of first category.

Almost weakly stable unitary groups are residual.

Results generalize classical theorems to operator semigroups.

Abstract

Inspired by the classical category theorems of Halmos and Rohlin for the discrete measure preserving transformations, we prove analogous results in the abstract setting of unitary and isometric C_0-semigroups on a separable Hilbert space. More presicely, we show that the set of all weakly stable unitary groups (isometric semigroups) is of first category, while the set of all almost weakly stable unitary groups (isometric semigroups) is residual for an appropriate topology.