On the complete metrisability of spaces of contractive semigroups

TL;DR

This paper proves that the space of contractive $C_0$-semigroups on a separable infinite-dimensional Hilbert space is a Polish space, resolving an open problem about its complete metrisability using Borel complexity and automatic continuity.

Contribution

It establishes that the space of contractive $C_0$-semigroups is Polish, providing a positive answer to the open problem about its complete metrisability.

Findings

The space of contractive $C_0$-semigroups is a Polish space.

Utilizes Borel complexity and automatic continuity to prove metrisability.

Addresses an open problem from Eisner (2010).

Abstract

The space of unitary $C_{0}$-semigroups on separable infinite dimensional Hilbert space, when viewed under the topology of uniform weak convergence on compact subsets of $\mathbb{R}_{+}$, is known to admit various interesting residual subspaces. Before treating the contractive case, the problem of the complete metrisability of this space was raised in [Eisner, 2010]. Utilising Borel complexity computations and automatic continuity results for semigroups, we obtain a general result, which in particular implies that the one-/multiparameter contractive $C_{0}$-semigroups constitute Polish spaces and thus positively addresses the open problem.