Characterisations of dilations via approximants, expectations, and functional calculi

TL;DR

This paper characterizes unitary dilations and approximations of classical dynamical systems on Hilbert spaces, extending known results to unbounded cases and distinguishing between different notions of weak convergence.

Contribution

It introduces new characterizations of unitary dilations via approximants and functional calculi, extending previous work to unbounded operators and non-commutative systems.

Findings

Characterizes simultaneous regular unitary dilatability using approximants.

Establishes a topological distinction between two notions of unitary dilations.

Applicable to both commutative and non-commutative systems satisfying CCR.

Abstract

We consider characterisations of unitary dilations and approximations of irreversible classical dynamical systems on a Hilbert space. In the commutative case, building on the work in [9], one can express well known approximants (e.g. Hille- and Yosida-approximants) via expectations over certain stochastic processes. Using this, our first result characterises the simultaneous regular unitary dilatability of commuting families of $C_{0}$-semigroups via the dilatability of such approximants as well as via regular polynomial bounds. This extends the results in [13] to the unbounded setting. We secondly consider characterisations of unitary and regular unitary dilations via two distinct functional calculi. Applying these tools to a large class of classical dynamical systems, these two notions of dilation exactly characterise when a system admits unitary approximations under certain distinct notions of weak convergence. This establishes a sharp topological distinction between the two notions of unitary dilations. Our results are applicable to commutative systems as well as non-commutative systems satisfying the canonical commutation relations (CCR) in the Weyl form.