On the complete metrisability of spaces of contractive semigroups

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.

Minor stylistic changes. Appendix included which provides a larger class of examples to which the main result applies. Note that this appendex consists of the content of the author's paper 'On the strong continuity of generalised semigroups' (arXiv:2009.04393)

References in corpus (3)

• Linear dynamical systems on Hilbert spaces: typical properties and explicit examples
• The space of contractive $C_0$-semigroups is a Baire space
• On the strong continuity of generalised semigroups

Cited by in corpus (3)

• Characterisations of dilations via approximants, expectations, and functional calculi
• Interpolation and non-dilatable families of ${\mathcal{C}}_0$-semigroups
• Free dilations of families of ${\mathcal{C}}_0$-semigroups and applications to evolution families