Unitary Colligations, Reproducing Kernel Hilbert Spaces, and Nevanlinna–Pick Interpolation in Several Variables

Ball, Joseph A.; Trent, Tavan T.

doi:10.1006/jfan.1998.3278

Cited by 133 publications

(134 citation statements)

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“…Remark 3.9 The lack of uniqueness of a closely connected conservative realization for a given function from the generalized Schur class is not a specific character of our approach: a similar phenomenon appears in [8]. where ζA (0) ∈ [X ⊖ X ′ , X ⊖ X ′ ] and ζG ′ ∈ [X ′ ⊕ N − , X ′ ⊕ N + ] (ζ ∈ T N ) are linear pencils of unitary operators, and we get X 1 ⊂ X ′ = X (see (3.15)) that contradicts to the close connectedness of α.…”

Section: Is Unitary and Represented By The Block-diagonal Matrixmentioning

confidence: 84%

“…Ball and T.T. Trent [8] systems known in literature on the multidimensional systems theory as the "Roesser model" (see e.g. [15]), with imposed metric constraints, namely conservative (in particular, unitary) systems are investigated (in the terminology of the authors such a system is called conservative if it satisfies one of "energy" equalities, and unitary if it satisfies both of these equalities analogous to (0.4)).…”

Section: Introductionmentioning

confidence: 99%

“…[15]), with imposed metric constraints, namely conservative (in particular, unitary) systems are investigated (in the terminology of the authors such a system is called conservative if it satisfies one of "energy" equalities, and unitary if it satisfies both of these equalities analogous to (0.4)). Using the result of [2] on the realization of operator-valued functions from the generalized Schur class (we shall recall its definition in Section 3) by unitary systems of the above-mentioned type, in [8] various functional models of those systems are constructed, and also interpolation problems for holomorphic functions on D N are solved. Our paper has the following structure.…”

Section: Introductionmentioning

confidence: 99%

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Multiparametric dissipative linear stationary dynamical scattering systems: Discrete case, II: Existence of conservative dilations

Kalyuzhniy

2000

Integr equ oper theory

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We propose the new generalization of linear stationary dynamical systems with discrete time t ∈ Z to the case t ∈ Z N . The dynamics of such a system can be reproduced by means of its associated multiparametric Lax-Phillips semigroup. We define multiparametric dissipative, and conservative scattering systems and interpret them in terms of operator colligations, of the associated semigroup and of "energy" relations for system data. We prove the Agler's type theorem describing the class of holomorphic operator-valued functions on the polydisc D N that are the transfer functions of multiparametric conservative scattering systems.

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Section: Is Unitary and Represented By The Block-diagonal Matrixmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Multiparametric dissipative linear stationary dynamical scattering systems: Discrete case, II: Existence of conservative dilations

Kalyuzhniy

2000

Integr equ oper theory

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“…Since the Cayley transform over the values of functions maps the class AH N (U) into the Agler-Schur class AS N (U) (see [1]), the composition of these two transforms (the double Cayley transform) maps B N (U) into AS N (U). The class AS N (U) is important in the interpolation theory and systems theory in several variables (see, e.g., [1,2,6,14,15,5]), that is why it is interesting to describe the image of B N (U) in AS N (U) under the double Cayley transform. Given f ∈ B N (U), define for w ∈ D N :…”

Section: The Image Of the Bessmertnyȋ Class Under The Double Cayley Tmentioning

confidence: 99%

On the Bessmertnyĭ Class of Homogeneous Positive Holomorphic Functions of Several Variables

Kalyuzhnyĭ-Verbovetzkiĭ

2004

Current Trends in Operator Theory and Its Applications

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Abstract. The class of operator-valued functions which are homogeneous of degree one, holomorphic in the open right polyhalfplane, have positive semidefinite real parts there and take selfadjoint operator values at real points, and its subclass consisting of functions representable in the form of Schur complement of a block of a linear pencil of operators with positive semidefinite operator coefficients, are investigated. The latter subclass is a generalization of the class of characteristic matrix functions of passive 2n-poles considered as functions of impedances of its elements, which was introduced by M. F. Bessmertnyȋ. Several equivalent characterizations of the generalized Bessmertnyȋ class are given, and its intimate connection with the Agler-Schur class of holomorphic contractive operator-valued functions on the unit polydisk is established.

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“…In [3] Ball and Trent prove a generalization of Leech's theorem to the polydisc in C d , adapting a technique coined the 'lurking isometry' approach in [2], and give a…”

Section: Introductionmentioning

confidence: 99%

Rational matrix solutions to the Leech equation: The Ball–Trent approach revisited

Horst

2013

Journal of Mathematical Analysis and Applications

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Abstract. Using spectral factorization techniques, a method is given by which rational matrix solutions to the Leech equation with rational matrix data can be computed explicitly. This method is based on an approach by J.A. Ball and T.T. Trent, and generalizes techniques from recent work of T.T. Trent for the case of polynomial matrix data. IntroductionConsider H ∞ -matrix functions G ∈ H K is positive. Note that (0.1) is equivalent to T G T X = T K and T X ≤ 1. Hence Leech's theorem can be viewed as the analogue of the Douglas factorization lemma [7] within the class of analytic Toeplitz operators. The necessity of (0.2) follows directly from Douglas' factorization lemma and the reformulation of (0.1) in terms of Toeplitz operators. The other implication is more involved. The solution criterion (0.2) can also be formulated directly in terms of the functions G and K, it is equivalent to the mapbeing a positive kernel in the sense of Aronszajn [1], that is, for any finite sequence z 1 , . . . , z n ∈ D the block operator matrix [L(z i , z j )] i,j=1,...,n defines a positive operator on the Hilbert space direct sum of n copies of C m . We note that the actual result by Leech is stated in the general context of Hilbert space operators intertwining shift operators, and in particular holds for operator-valued H ∞ -functions as well. Our interest is primarily in the case where G and K are rational matrix functions.There exists various proofs of Leech's theorem, see [10] and the references therein. In [3] Ball and Trent prove a generalization of Leech's theorem to the polydisc in C d , adapting a technique coined the 'lurking isometry' approach in [2], and give aKey words and phrases. Leech equation, Toeplitz operators, stable rational matrix functions, outer spectral factor. 1 2 SANNE TER HORST description of all X ∈ H ∞ p×q satisfying (0.1). We briefly outline the construction here, specified to the single variable case.The positivity of T G T *. Such a factorization is often referred to as a Kolmogorov decomposition in the literature, cf., [6]. LetΛ • be the analytic operator-valued function on D, with valuesΛ From this identity one derives the existence of a partial isometryThis in turn implies that the function X defined on D byis in H From the point of view of rational matrix functions the above construction has one disadvantage. In general, the Hilbert space H • appearing in (0.5) is infinite dimensional, and in that case it is hard to see when the solution X in (0.7) is rational. In fact, even if both G and K are rational matrix functions,may very well be of infinite rank. More precisely, see Theorem 3.2 below, in the rational matrix case* for all 0 ≤ t ≤ 2π. Overcoming this difficulty is the main theme of the present paper.In the context of the Toeplitz-corona problem, which can be reduced to the special case of (0.1) with q = m and K(z) = I m , z ∈ D, Trent [12] deduced a modification of the above procedure for the special case that G is a row vector (m = 1) polynomial, leading to a rational column vector sol...

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Unitary Colligations, Reproducing Kernel Hilbert Spaces, and Nevanlinna–Pick Interpolation in Several Variables

Cited by 133 publications

References 19 publications

Multiparametric dissipative linear stationary dynamical scattering systems: Discrete case, II: Existence of conservative dilations

Multiparametric dissipative linear stationary dynamical scattering systems: Discrete case, II: Existence of conservative dilations

On the Bessmertnyĭ Class of Homogeneous Positive Holomorphic Functions of Several Variables

Rational matrix solutions to the Leech equation: The Ball–Trent approach revisited

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