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 inde-terminates with norm-square-summable vector coefficients, and (2) the reproducing kernel Hilbert space (often now called the Arveson space) over the unit ball in C{sup}d with reproducing kernel k(λ,ζ) - 1/(1 - <λ,ζ>) (λμ ∈ C{sup}d with ||λ||, ||ζ|| ＜ 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) Foraasini-Marchesini-type system with evolution along the integer lattice Z{sup}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.
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