Title page for ETD etd-05102012-184739

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
  • 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
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×) which together form a direct-sum decomposition of L2. In the case where the pair (M,M×) 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×). 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×) 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.
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