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

TL;DR

This paper extends interpolation techniques for commuting contraction families to $ ext{C}_0$-semigroups, constructs counterexamples for unitary dilations in higher dimensions, and explores implications for embedding and rigidity problems in operator theory.

Contribution

It generalizes interpolation methods to $ ext{C}_0$-semigroups, constructs non-dilatable examples, and advances understanding of embedding and rigidity in operator families.

Findings

Existence of non-unitarily dilatable $ ext{C}_0$-semigroup families for $d \\geq 3$

Residuality of non-dilatable semigroups in the topology considered

Extension of embedding results for typical pairs of commuting operators

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 $d \geq 3$ 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 $\mathbb{R}_{\geq 0}^{d}$ of non-unitarily dilatable and non-unitarily approximable $d$-parameter contractive $\mathcal{C}_{0}$-semigroups on separable infinite-dimensional Hilbert spaces for each $d \geq 3$. 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).