Given a full-range simply-invariant shift-invariant subspace <i>M</i> of the vector-valued <i>L<sup>2</sup></i> space on the unit circle, the classical Beurling-Lax-Halmos (BLH) theorem obtains a unitary operator-valued function <i>W</i> so that <i>M</i> may be represented as the image of of the Hardy space <i>H<sup>2</sup></i> on the disc under multiplication by <i>W</i>. The work of Ball-Helton later extended this result to find a single function representing a so-called dual shift-invariant pair of subspaces <i>(M,M<sup>Ã </sup>)</i> which together form a direct-sum decomposition of <i>L<sup>2</sup></i>. In the case where the pair <i>(M,M<sup>Ã </sup>)</i> are finite-dimensional perturbations of the Hardy space <i>H<sup>2</sup></i> and its orthogonal complement, Ball-Gohberg-Rodman obtained a transfer function realization for the representing function <i>W</i>; this realization was parameterized in terms of zero-pole data computed from the pair <i>(M,M<sup>Ã </sup>)</i>. Later work by Ball-Raney extended this analysis to the case of nonrational functions <i>W</i> 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 <i>(M,M<sup>Ã </sup>)</i> of the <i>L<sup>2</sup></i> 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. / Ph. D.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/27636 |
| Date | 30 May 2012 |
| Creators | Amaya, Austin J. |
| Contributors | Mathematics, Ball, Joseph A., Hagedorn, George A., Klaus, Martin, Renardy, Michael J. |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Detected Language | English |
| Type | Dissertation |
| Format | application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
| Relation | Amaya_AJ_D_2012.pdf |
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