Multivariable backward-shift-invariant subspaces and observability operators

Abstract

It is well known that subspaces of the Hardy space over the unit disk which are invariant under the backward shift occur as the image of an observability operator associated with a discrete-time linear system with stable state-dynamics, as well as the functional-model space for a Hilbert space contraction operator. We discuss two multivariable extensions of this structure, where the classical Hardy space is replaced by (1) the Fock space of formal power series in a collection of d noncommuting indeterminates with norm-square-summable vector coefficients, and (2) the reproducing kernel Hilbert space (often now called the Arveson space) over the unit ball in ${\mathbb{C}^{d}}$ with reproducing kernel ${k(\lambda, \zeta) = 1/(1 - \langle \lambda, \zeta \rangle) (\lambda, \zeta \in \mathbb{C}^{d} with \| \lambda \|, \| \zeta \| < 1}$). In the first case, the associated linear system is of noncommutative Fornasini–Marchesini type with evolution along a free semigroup with d generators, while in the second case the linear system is a standard (commutative) Fornasini–Marchesini-type system with evolution along the integer lattice ${\mathbb{Z}^{d}}$ . An abelianization map (or symmetrization of the Fock space) links the first case with the second. The second case has special features depending on whether the operator-tuple defining the state dynamics is commutative or not. The paper focuses on multidimensional state-output linear systems and the associated observability operators; followup papers Ball, Bollotnikov, and Fang (2007a, 2007b) use the results here to extend the analysis to represent observability-operator ranges as reproducing kernel Hilbert spaces with reproducing kernels constructed from the transfer function of a conservative multidimensional (noncommutative or commutative) input-state-output linear system.