Discrete-Time Dichotomous Well-Posed Linear Systems and Generalized Schur–Nevanlinna–Pick Interpolation

Ball, Joseph A.; Raney, Michael

doi:10.1007/s11785-006-0001-y

Cited by 21 publications

(20 citation statements)

References 31 publications

(25 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the single-variable case (d = 1), we have J = J and these functions coincide with the strongly regular J-inner functions in the sense of Arov-Dym [9] (see also [24]) which are analytic on the unit disk D = B 1 .…”

Section: Theorem 24 Suppose That (C T) Is An Analytic Output-pairmentioning

confidence: 64%

Interpolation Problems for Schur Multipliers on the Drury-Arveson Space: from Nevanlinna-Pick to Abstract Interpolation Problem

Ball

Bolotnikov

2008

Integr. equ. oper. theory

Self Cite

View full text Add to dashboard Cite

We survey various increasingly more general operator-theoretic formulations of generalized left-tangential Nevanlinna-Pick interpolation for Schur multipliers on the Drury-Arveson space. An adaptation of the methods of Potapov and Dym leads to a chain-matrix linear-fractional parametrization for the set of all solutions for all but the last of the formulations for the case where the Pick operator is invertible. The last formulation is a multivariable analogue of the Abstract Interpolation Problem formulated by Katsnelson, Kheifets and Yuditskii for the single-variable case; we obtain a Redheffertype linear-fractional parametrization for the set of all solutions (including in degenerate cases) via an adaptation of ideas of Arov and Grossman. Mathematics Subject Classification (2000). Primary 47A57; Secondary 47A48, 47B32, 47B50, 32A05.Keywords. Abstract Interpolation Problem, linear-fractional map, Stein equation, left tangential interpolation with operator argument.

show abstract

Section: Theorem 24 Suppose That (C T) Is An Analytic Output-pairmentioning

confidence: 64%

Interpolation Problems for Schur Multipliers on the Drury-Arveson Space: from Nevanlinna-Pick to Abstract Interpolation Problem

Ball

Bolotnikov

2008

Integr. equ. oper. theory

Self Cite

View full text Add to dashboard Cite

show abstract

“…We shall call any such S a representer of M. Here we present a realization-theoretic proof of the H Y (k d )-Beurling-Lax theorem as an application of Theorem 4.2 (see [9,10] for an illustration of this approach for the case d = 1). We first need some preliminaries.…”

Section: Proof Let S ∈ S D (U Y) Be Inner Letmentioning

confidence: 99%

“…The idea in this approach is to represent the shift-invariant subspace M as the set of all H Y (k d )-solutions of fairly general set of homogeneous interpolation conditions, and then to construct a realization U = A B C D for S(λ) from the operators defining the homogeneous interpolation conditions. For the case d = 1, this approach can be found in [9] for the rational case and in [10] for the non-rational case, done there in the more complicated context where the shift-invariant subspace M is merely contained in the Y-valued L 2 space over the unit circle T and is not necessarily contained in the Hardy space…”

Section: Introductionmentioning

confidence: 99%

Schur-class multipliers on the Arveson space: De Branges–Rovnyak reproducing kernel spaces and commutative transfer-function realizations

Ball

Bolotnikov

Fang

2008

Journal of Mathematical Analysis and Applications

Self Cite

View full text Add to dashboard Cite

An interesting and recently much studied generalization of the classical Schur class is the class of contractive operator-valued multipliers S(λ) for the reproducing kernel Hilbert spaceThe reproducing kernel space H(K S ) associated with the positive kernel K S (λ, ζ ) = (I − S(λ)S(ζ ) * ) · k d (λ, ζ ) is a natural multivariable generalization of the classical de Branges-Rovnyak canonical model space. A special feature appearing in the multivariable case is that the space H(K S ) in general may not be invariant under the adjoints M * λ j of the multiplication operators M λ j : f (λ) → λ j f (λ) on H(k d ). We show that invariance of H(K S ) under M * λ j for each j = 1, . . . , d is equivalent to the existence of a realization for S(λ) of the form S(λ) = D + C⎦ has adjoint U * which is isometric on a certain natural subspace (U is "weakly coisometric") and has the additional property that the state operators A 1 , . . . , A d pairwise commute; in this case one can take the state space to be the functional-model space H(K S ) and the state operators A 1 , . . . , A d to be given by A j = M * λ j | H(K S ) (a de BrangesRovnyak functional-model realization). We show that this special situation always occurs for the case of inner functions S (where the associated multiplication operator M S is a partial isometry), and that inner multipliers are characterized by the existence of such a realization such that the state operators A 1 , . . . , A d satisfy an additional stability property.

show abstract

“…Given any k in a C * -algebra, find f , a strictly contractive extension for k (see Section 2 for the precise definition). The methods for solving a variety of extension problems has evolved into increasing levels of sophistication over the past two decades (see [3] for some recent results and a short overview). The article [7] enhances the Grassmannian version of the band method to handle the NehariTakagi problem rather than merely the Nehari problem.…”

Section: Introductionmentioning

confidence: 99%

On Some Aspects of the J-Spectral Factorization

Iftime¹

2011

Annals of the Alexandru Ioan Cuza University - Mathematics

View full text Add to dashboard Cite

Abstract. The J-spectral factorization problem naturally arises in control theory and it plays an important role in H∞-control, linear quadratic optimal control, Hankel norm approximation problem. Characterization of solution for control problems is sometimes given using the J-spectral factor(s). The paper has the nature of a survey article. We review the band method version of the Grassmannian approach for solving strictly contractive extension problems and the J-spectral factorization approach for solving the suboptimal Nehari problem in the setting of the Wiener algebra on the imaginary axis. The new (and modest) contributions is to clarify the connections between the two approaches.Mathematics Subject Classification 2000: 47N70, 93A99.

show abstract

Discrete-Time Dichotomous Well-Posed Linear Systems and Generalized Schur–Nevanlinna–Pick Interpolation

Cited by 21 publications

References 31 publications

Interpolation Problems for Schur Multipliers on the Drury-Arveson Space: from Nevanlinna-Pick to Abstract Interpolation Problem

Interpolation Problems for Schur Multipliers on the Drury-Arveson Space: from Nevanlinna-Pick to Abstract Interpolation Problem

Schur-class multipliers on the Arveson space: De Branges–Rovnyak reproducing kernel spaces and commutative transfer-function realizations

On Some Aspects of the J-Spectral Factorization

Contact Info

Product

Resources

About