Interpolation and non-dilatable families of $$\mathcal {C}_{0}$$ -semigroups

Abstract

We generalise a technique of Bhat and Skeide (J Funct Anal 269:1539–1562, 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 \(\textsc {wot}\)-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, viz. that ‘typical’ pairs of commuting operators can be simultaneously embedded into commuting pairs of \(\mathcal {C}_{0}\)-semigroups, which extends results of Eisner (2009–2010).