The space of contractive $C_{0}$-semigroups is a Baire space

TL;DR

This paper proves that the space of contractive $C_{0}$-semigroups on infinite dimensional Hilbert spaces is a Baire space by transferring topological properties from dense subspaces, using game-theoretic methods.

Contribution

It establishes that the space of contractive $C_{0}$-semigroups is a Baire space, solving an open problem by leveraging topological transfer techniques and classification via infinite games.

Findings

The space of unitary semigroups is completely metrisable.

Certain topological properties can be transferred from dense subspaces.

The approach applies to other contexts like contractions under pw-topology.

Abstract

Working over infinite dimensional separable Hilbert spaces, residual results have been achieved for the space of contractive $C_{0}$-semigroups under the topology of uniform weak operator convergence on compact subsets of $\mathbb{R}_{+}$. Eisner and Ser\'eny raised in 2009 the open problem: Does this space constitute a Baire space? Observing that the subspace of unitary semigroups is completely metrisable and appealing to known density results, we solve this problem positively by showing that certain topological properties can in general be transferred from dense subspaces to larger spaces. The transfer result in turn relies upon classification of topological properties via infinite games. Our approach is sufficiently general and can be applied to other contexts, e.g. the space of contractions under the pw-topology.