Free dilations of families of $\mathcal{C}_{0}$-semigroups and applications to evolution families

Abstract

Commuting families of contractions or contractive $\mathcal{C}_{0}$-semigroups on Hilbert spaces often fail to admit power dilations resp, simultaneous unitary dilations which are themselves commutative (see [45, 13, 15]). In the \emph{non-commutative} setting, Sz.-Nagy [60] and Bo\.{z}ejko [5] provided means to dilate arbitrary families of contractions. The present work extends these discrete-time results to families $\{T_{i}\}_{i \in I}$ of contractive $\mathcal{C}_{0}$-semigroups. We refer to these dilations as continuous-time \emph{free unitary dilations} and present three distinct approaches to obtain them: 1) An explicit derivation applicable to semigroups that arise as interpolations; 2) A full proof with an explicit construction, via the theory of co-generators \`{a} la S{\l}oci\'{n}ski [54, 55]; and 3) A second full proof based on the abstract structure of semigroups, which admits a natural reformulation to semigroups defined over topological free products of $\mathbb{R}_{\geq 0}$ and leads to various residuality results. In 2) a II\textsuperscript{nd} free dilation theorem for topologised index sets is developed via a reformulation of the Trotter--Kato theorem for co-generators. As an application of this we demonstrate how evolution families can be reduced to continuously monitored processes subject to temporal change, \`{a} la the quantum Zeno effect [22, 23, 24, 30, 37].