2009/06/04 by Joseph A. Ball, J. A. Ball, Ball, J. A. +2
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Quantum Mechanics and Applications #Spectral Theory in Mathematical Physics #math.FA #math.OA
paper · pdf · doi:10.48550/arxiv.0906.0988
To appear in proceedings of the nDS 09 conference in Thessaloniki, Greece
arxiv created 2009/06/04 · arxiv updated 2009/12/01
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 Cd, 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.