Multivariable backward-shift-invariant subspaces and observability operators
| dc.creator | Ball, Joseph A. | |
| dc.creator | Bolotnikov, Vladimir | |
| dc.creator | Fang, Quanlei | |
| dc.date | 2006-10-20 | |
| dc.date.accessioned | 2026-07-07T07:29:16Z | |
| dc.date.available | 2026-07-07T07:29:16Z | |
| dc.description | It is well known that subspaces of the Hardy space over the unit disk which are invariant under the backward shift occur as the image of an observability operator associated with a discrete-time linear system with stable state-dynamics, as well as the functional-model space for a Hilbert space contraction operator. We discuss two multivariable extensions of this structure, where the classical Hardy space is replaced by (1) the Fock space of formal power series in a collection of $d$ noncommuting indeterminates with norm-square-summable vector coefficients, and (2) the reproducing kernel Hilbert space (often now called the Arveson space) over the unit ball in ${\mathbb C}^{d}$ with reproducing kernel $k(λ, ζ) = 1/(1 - < λ, ζ>)$ ($\lam, ζ\in {\mathbb C}^{d}$ with $\| λ\|, \| ζ\| < 1$). In the first case, the associated linear system is of noncommutative Fornasini-Marchesini type with evolution along a free semigroup with $d$ generators, while in the second case the linear system is a standard (commutative) Fornasini-Marchesini-type system with evolution along the integer lattice ${\mathbb Z}^{d}$. An abelianization map (or symmetrization of the Fock space) links the first case with the second. The second case has special features depending on whether the operator-tuple defining the state dynamics is commutative or not. The paper focuses on multidimensional state-output linear systems and the associated observability operators. | |
| dc.identifier | https://arxiv.org/abs/math/0610634 | |
| dc.identifier | http://arxiv.org/abs/math/0610634 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/118042 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | Commutative Algebra | |
| dc.subject | 47A57 | |
| dc.title | Multivariable backward-shift-invariant subspaces and observability operators | |
| dc.type | text |