Dilations of commuting $C_{0}$-semigroups with bounded generators and the von Neumann polynomial inequality

Abstract

Consider $d$ commuting $C_{0}$-semigroups (or equivalently: $d$-parameter $C_{0}$-semigroups) over a Hilbert space for $d \in \mathbb{N}$. In the literature (\textit{cf.} [29, 26, 27, 23, 18, 25]), conditions are provided to classify the existence of unitary and regular unitary dilations. Some of these conditions require inspecting values of the semigroups, some provide only sufficient conditions, and others involve verifying sophisticated properties of the generators. By focussing on semigroups with bounded generators, we establish a simple and natural condition on the generators, \textit{viz.} \emph{complete dissipativity}, which naturally extends the basic notion of the dissipativity of the generators. Using examples of non-doubly commuting semigroups, this property can be shown to be strictly stronger than dissipativity. As the first main result, we demonstrate that complete dissipativity completely characterises the existence of regular unitary dilations, and extend this to the case of arbitrarily many commuting $C_{0}$-semigroups. We furthermore show that all multi-parameter $C_{0}$-semigroups (with bounded generators) admit a weaker notion of regular unitary dilations, and provide simple sufficient norm criteria for complete dissipativity. The paper concludes with an application to the von Neumann polynomial inequality problem, which we formulate for the semigroup setting and solve negatively for all $d \geq 2$.