A W*-correspondence approach to multi-dimensional linear dissipative systems

TL;DR

This paper explores a mathematical framework based on W*-correspondences to analyze multi-dimensional linear dissipative systems, connecting operator algebra theory with system theory concepts like transfer functions and Z-transforms.

Contribution

It introduces a W*-correspondence approach to multi-dimensional dissipative systems, extending the Muhly-Solel formalism to specific classes of such systems.

Findings

Unified operator algebra framework for multi-dimensional systems

Characterization of transfer functions as holomorphic operator-valued functions

Application to both commutative and noncommutative dissipative systems

Abstract

Recent work of the operator algebraists P. Muhly and B. Solel, primarily motivated by the theory of operator algebras and mathematical physics, delineates a general abstract framework where system theory ideas appear in disguised form. These system-theory ingredients include: system matrix for an input/state/output linear system, Z-transform from a "time domain" to a "frequency domain", and Z-transform of the output signal given by an observation function applied to the initial condition plus a transfer function applied to the Z-transform of the input signal. Here we set down the definitions and main results for the general Muhly-Solel formalism and illustrate them for two specific types of multi-dimensional linear systems: (1) dissipative Fornasini-Marchesini state-space representations with transfer function equal to a holomorphic operator-valued function on the unit ball in ${\mathbb C}^d$, and (2) noncommutative dissipative Fornasini-Marchsini linear systems with evolution along a free semigroup and with transfer function defined on the noncommutative ball of strict row contractions on a Hilbert space.