On the complete metrisability of spaces of contractive semigroups

Abstract

AbstractThe space of unitary$$C_{0}$$C0-semigroups on a separable infinite-dimensional Hilbert space, when viewed under the topology of uniform weak operator convergence on compact subsets of$${\mathbb {R}}_{+}$$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 [4]. Utilising Borel complexity computations and automatic continuity results for semigroups, we obtain a general result, which in particular implies that the one-/multiparameter contractive$$C_{0}$$C0-semigroups constitute Polish spaces and thus positively addresses the open problem.