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 <i>M</i> of the vector-valued <i>L²</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>H²</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^Ã)</i> which together form a direct-sum decomposition of <i>L²</i>. In the case where the pair <i>(M,M^Ã)</i> are finite-dimensional perturbations of the Hardy space <i>H²</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^Ã)</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^Ã)</i> of the <i>L²</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.

Name thesis:degree_name

Ph. D.

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Mathematics

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Virginia Tech

Year dc:date.issued

2012

Author dc:creator

• Amaya, Austin J.

Chair dc:contributor.committeechair

• Ball, Joseph A.

Committee members dc:contributor.committeemember

• Hagedorn, George A.
• Klaus, Martin
• Renardy, Michael J.

• 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

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Dc Identifier Other

etd-05102012-184739

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/27636