Beurling-Lax Representations of Shift-Invariant Spaces, Zero-Pole Data Interpolation, and Dichotomous Transfer Function Realizations: Half-Plane/Continuous-Time Versions

Given a full-range simply-invariant shift-invariant subspace M of the vector-valued L² space on the unit circle, the classical Beurling-Lax-Halmos (BLH) theorem obtains a unitary operator-valued function W so that M may be represented as the image of of the Hardy space H² on the disc under multiplication by W. The work of Ball-Helton later extended this result to find a single function representing a so-called dual shift-invariant pair of subspaces (M,M^Ã ) which together form a direct-sum decomposition of L². In the case where the pair (M,M^Ã ) are finite-dimensional perturbations of the Hardy space H² and its orthogonal complement, Ball-Gohberg-Rodman obtained a transfer function realization for the representing function W; this realization was parameterized in terms of zero-pole data computed from the pair (M,M^Ã ). Later work by Ball-Raney extended this analysis to the case of nonrational functions W where the zero-pole data is taken in an infinite-dimensional operator theoretic sense. The current work obtains analogues of these various results for arbitrary dual shift-invariant pairs (M,M^Ã ) of the L² spaces on the real line; here, shift-invariance refers to invariance under the translation group. These new results rely on recent advances in the understanding of continuous-time infinite-dimensional input-state-output linear systems which have been codified in the book by Staffans.