On the complete metrisability of spaces of contractive semigroups

DOI10.1007/S00013-022-01717-1arXiv2010.01573MaRDI QIDQ6350491

Raj Dahya

Publication date: 4 October 2020

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.

Mathematics Subject Classification ID

Metric spaces, metrizability (54E35) One-parameter semigroups and linear evolution equations (47D06)

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