Dilations of Semigroups of Contractions through Vessels

TL;DR

This paper introduces a method to construct dilations of multi-parameter contraction semigroups generated by commuting dissipative operators using the theory of operator vessels, advancing the understanding of operator dilations.

Contribution

It provides a novel dilation construction for multi-parameter semigroups of contractions via operator vessels, connecting system theory with operator dilation theory.

Findings

Constructs dilations for multi-parameter contraction semigroups

Uses operator vessel theory to achieve dilation

Bridges system theory and operator dilation concepts

Abstract

Let $A_1,\ldots,A_d$ be a $d$-tuple of commuting dissipative operators on a separable Hilbert space $\mathcal{H}$. Using the theory of operator vessels and their associated systems, we give a construction of a dilation of the multi-parameter semigroup of contractions on $\mathcal{H}$ given by $e^{i \sum_{j=1}^d t_j A_j}$.