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

##### Abstract

Given a full-range simply-invariant shift-invariant subspace

*M*of the vector-valued*L*space on the unit circle, the classical Beurling-Lax-Halmos (BLH) theorem obtains a unitary operator-valued function^{2}*W*so that*M*may be represented as the image of of the Hardy space*H*on the disc under multiplication by^{2}*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*which together form a direct-sum decomposition of^{Ã })*L*. In the case where the pair^{2}*(M,M*are finite-dimensional perturbations of the Hardy space^{Ã })*H*and its orthogonal complement, Ball-Gohberg-Rodman obtained a transfer function realization for the representing function^{2}*W*; this realization was parameterized in terms of zero-pole data computed from the pair*(M,M*. Later work by Ball-Raney extended this analysis to the case of nonrational functions^{Ã })*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*of the^{Ã })*L*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.^{2}##### Collections

- Doctoral Dissertations [11228]