Canonical Agler decompositions and transfer function realizations

Bickel, Kelly; Knese, Greg

doi:10.1090/tran/6542

Cited by 10 publications

(11 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Also note that, in the above theorem, one can choose the coefficient Hilbert space D as ran(I − T T * ) (see [13]). In contrast with the von-Neumann and Wold decomposition theorem for isometries, the structure of commuting n-tuples of isometries, n ≥ 2, is much more complicated and very little, in general, is known (see [3,4,5,6,7,11,12,17,18,15]). However, for pure pairs of commuting isometries, the problem is more tractable.…”

Section: Introductionmentioning

confidence: 99%

Factorizations of contractions

Das

Sarkar

2017

Advances in Mathematics

View full text Add to dashboard Cite

The celebrated Sz.-Nagy and Foias theorem asserts that every pure contraction is unitarily equivalent to an operator of the form, for some Hilbert space D. On the other hand, the celebrated theorem of Berger, Coburn and Lebow on pairs of commuting isometries can be formulated as follows: a pure isometry V on a Hilbert space H is a product of two commuting isometries V 1 and V 2 in B(H) if and only if there exist a Hilbert space E, a unitary U in B(E) and an orthogonal projectionIn this context, it is natural to ask whether similar factorization results hold true for pure contractions. The purpose of this paper is to answer this question. More particularly, let T be a pure contraction on a Hilbert space H and let P Q M z | Q be the Sz.-Nagy and Foias representation of T for some canonical Q ⊆ H 2 D (D). Then T = T 1 T 2 , for some commuting contractions T 1 and T 2 on H, if and only if there exist B(D)-valued polynomials ϕ and ψ of degree ≤ 1 such that Q is a joint (M * ϕ , M * ψ )-invariant subspace,Moreover, there exist a Hilbert space E and an isometry V ∈ B(D; E) such thatwhere the pair (Φ, Ψ), as defined above, is the Berger, Coburn and Lebow representation of a pure pair of commuting isometries on H 2 E (D). As an application, we obtain a sharper von Neumann inequality for commuting pairs of contractions.2010 Mathematics Subject Classification. 47A13, 47A20, 47A56, 47A68, 47B38, 46E20, 30H10.

show abstract

Section: Introductionmentioning

confidence: 99%

Factorizations of contractions

Das

Sarkar

2017

Advances in Mathematics

View full text Add to dashboard Cite

show abstract

“…satisfying the left-and right interpolation conditions G(λ Remark 7 (Operator-theoretic approaches to bivariate interpolation). Agler, McCarthy and others worked on discrete nD metric interpolation problems (see [1,4,6]) using operator-theoretic techniques, see also [5]. Interesting similarities exist between their formulas and ours, compare e.g.…”

Section: Remark 5 (Data Dualization Via Mirroring)mentioning

confidence: 88%

State-Space Modeling of Two-Dimensional Vector-Exponential Trajectories

Rapisarda¹,

Antoulas²

2016

SIAM J. Control Optim.

View full text Add to dashboard Cite

Abstract. We solve two problems in modelling polynomial vector-exponential trajectories dependent on two independent variables. In the first one we assume that the data-generating system has no inputs, and we compute a state representation of the Most Powerful Unfalsified Model for this data. In the second instance we assume that the data-generating system is controllable and quarterplane causal, and we compute a Roesser i/s/o model. We provide procedures for solving these identification problems, both based on the factorization of constant matrices directly constructed from the data, from which state trajectories can be computed.Key words. multidimensional systems, most powerful unfalsified model, Roesser models, bilinear differential forms AMS subject classifications. 93A30, 93B15, 93B20, 93B30, 93C201. Introduction. We consider two problems in modelling two-dimensional (2D in the following) continuous trajectories from data. In both cases the data consistis of polynomial vector-exponential trajectories, and we seek state-space models explaining it, i.e. systems of partial differential equations of first order in an auxiliary, "state" variable, and zeroth-order in the measured, "external" variable. The two situations differ in the model class we assume the data-generating system belongs to: in the first case we seek an autonomous state model , i.e. a system without inputs; in the second one we assume that an input/output partition of the external variable is given, and we compute an input-state-output (i/s/o in the following) model.Modelling 2D polynomial vector-exponential trajectories with autonomous systems has been considered in [32,33], on whose results the first part of this paper on the computation of autonomous models heavily relies. Modelling vector-exponential trajectories with transfer-function (i.e. input-output) models is closely related to twovariable rational interpolation; the latter has been investigated in the SISO case in [2]. The approach taken in the present paper differs fundamentally from those: we use data to first compute state trajectories corresponding to it, and in a second stage we compute a state representation for the data and the identified state trajectories by solving linear equations in the unknown state-, input-and output matrices.Modelling methodologies where state-trajectories are computed from data and state-equations are subsequently computed are well-known in the 1D case as subspace identification methods (see e.g. [12]). Such ideas have been pursued much less frequently in the 2D case: see [7,19] for a pioneering subspace-identification approach to the computation of i/s/o representations of denominator-separable 2D discretesystems from data. Our modelling approach differs essentially from those, in that it does not exploit the shift-invariance properties of data trajectories, but rather the fact that external properties-i.e. properties at the level of external variables, in our case duality-are reflected into internal properties-i.e. at the level of state. To make the...

show abstract

“…(These vector polynomials are closely related to so-called Agler kernels for more general bounded analytic functions. See [1].) Lemma 7.3 of [10] proves that the values of (| A 1 |, | A 2 |) on T 2 are the same for all Agler pairs.…”

Section: Integrability Of Perturbationsmentioning

confidence: 99%

Extreme points and saturated polynomials

Knese¹

2019

Illinois J. Math.

Self Cite

View full text Add to dashboard Cite

We consider the problem of characterizing the extreme points of the set of analytic functions f on the bidisk with positive real part and f (0) = 1. If one restricts to those f whose Cayley transform is a rational inner function, one gets a more tractable problem. We construct families of such f that are extreme points and conjecture that these are all such extreme points. These extreme points are constructed from polynomials dubbed T 2 -saturated, which roughly speaking means they have no zeros in the bidisk and as many zeros as possible on the boundary without having infinitely many zeros.

show abstract

Canonical Agler decompositions and transfer function realizations

Cited by 10 publications

References 31 publications

Factorizations of contractions

Factorizations of contractions

State-Space Modeling of Two-Dimensional Vector-Exponential Trajectories

Extreme points and saturated polynomials

Contact Info

Product

Resources

About