Interpolation and non-dilatable families of ${\mathcal{C}}_0$-semigroups

Abstract

We generalise a technique of Bhat and Skeide (2015) to interpolate commuting families $\{S_i{\}}_{i\in \mathcal{I}}$ of contractions on a Hilbert space $\mathcal{H}$, to commuting families $\{T_i{\}}_{i\in \mathcal{I}}$ of contractive ${\mathcal{C}}_0$-semigroups on $L^2({\prod }_{i\in \mathcal{I}}\mathbb{T})\otimes \mathcal{H}$. As an excursus, we provide applications of the interpolations to time-discretisation and the embedding problem. Applied to Parrott's construction (1970), we then demonstrate for $d\in \mathbb{N}$ with $d\ge 3$ the existence of commuting families $\{T_i{\}}_{i=1}^d$ of contractive ${\mathcal{C}}_0$-semigroups which admit no simultaneous unitary dilation. As an application of these counter-examples, we obtain the residuality wrt. the topology of uniform wot-convergence on compact subsets of ${\mathbb{R}}_{\ge 0}^d$ of non-unitarily dilatable and non-unitarily approximable $d$-parameter contractive ${\mathcal{C}}_0$-semigroups on separable infinite-dimensional Hilbert spaces for each $d\ge 3$. Similar results are also developed for $d$-tuples of commuting contractions. And by building on the counter-examples of Varopoulos--Kaijser (1973--74), a 0--1-result is obtained for the von Neumann inequality. Finally, we discuss applications to rigidity as well as the embedding problem, \textit{viz.} that `typical' pairs of commuting operators can be simultaneously embedded into commuting pairs of ${\mathcal{C}}_0$-semigroups, which extends results of Eisner (2009--10).

Minor corrections to dichotomy statements in Lemma 3.4, Prop 3.5, Rem 3.10. Extended two applications: PW-density of embeddable $d$-tuples (§2.3, p.12, starting from paragraph before Prop 2.9) and recovery of $d$-parameter semigroups via time-discretisations (§2.4.2, pp.16-18). These extensions are currently not set to appear in the published version, though may occur later as addenda

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Cited by in corpus (1)

• Free dilations of families of ${\mathcal{C}}_0$-semigroups and applications to evolution families