Type of Document Dissertation Author Amaya, Austin J. Author's Email Address amaya@vt.edu URN etd-05102012-184739 Title Beurling-Lax Representations of Shift-Invariant Spaces, Zero-Pole Data Interpolation, and Dichotomous Transfer Function Realizations: Half-Plane/Continuous-Time Versions Degree PhD Department Mathematics Advisory Committee

Advisor Name Title Ball, Joseph A. Committee Chair Hagedorn, George A. Committee Member Klaus, Martin Committee Member Renardy, Michael J. Committee Member Keywords

- reproducing kernel Hilbert spaces
- Hardy spaces over left/right half plane
- admissible Sylvester data set
- operator Sylvester equation
- infinite dimensional zero-pole data
- continuous shift semigroups
- Ltwo well-posed linear systems
- continuous-time linear systems
Date of Defense 2012-04-26 Availability unrestricted AbstractGiven a full-range simply-invariant shift-invariant subspaceMof the vector-valuedLspace on the unit circle, the classical Beurling-Lax-Halmos (BLH) theorem obtains a unitary operator-valued function^{2}Wso thatMmay be represented as the image of of the Hardy spaceHon 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,Mwhich together form a direct-sum decomposition of^{×})L. In the case where the pair^{2}(M,Mare finite-dimensional perturbations of the Hardy space^{×})Hand 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^{×})Wwhere 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,Mof the^{×})Lspaces 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}Files

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