Functional Models and Minimal Contractive Liftings (original) (raw)
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Indiana University Mathematics Journal, 2012
We develop a d-variable analog of the two-component de Branges-Rovnyak reproducing kernel Hilbert space associated with a Schur-class function on the unit disk. In this generalization, the unit disk is replaced by the unit ball in d-dimensional complex Euclidean space, and the Schur class becomes the class of contractive multipliers on the Drury-Arveson space over the ball. We also develop some results on a model theory for commutative row contractions which are not necessarily completely noncoisometric (the case considered in earlier work of Bhattacharyya, Eschmeier and Sarkar).
A rich structure related to the construction of analytic matrix functions
Journal of Functional Analysis
We study certain interpolation problems for analytic 2 × 2 matrix-valued functions on the unit disc. We obtain a new solvability criterion for one such problem, a special case of the µ-synthesis problem from robust control theory. For certain domains X in C 2 and C 3 we describe a rich structure of interconnections between four objects: the set of analytic functions from the disc into X , the 2 × 2 matricial Schur class, the Schur class of the bidisc, and the set of pairs of positive kernels on the bidisc subject to a boundedness condition. This rich structure combines with the classical realisation formula and Hilbert space models in the sense of Agler to give an effective method for the construction of the required interpolating functions.
2014
We obtain in this paper a relation between the matrices of coefficients ~f Darlinton realizations of a j-expansive matrix-valued function T(z) and the contractive matrix-valued function. S(z), given, in. terms of T(z) as a linear fractional transformation, over T(z), with constant coefficients. 1. We recall sorne known results on contractive and j-expansive 'matrixvalued functions [1,2,3]. A matrix S is called contractive if 1 S*S ~ O, where 1 is a unit ma trix and the symbol * denotes Hermitian conjugation. Let ,J be a matrix for which J* = J and J2 = l. A matrix A is cal1ed J-expansive if A*JA J ~ O, and J-unitary if A*JA J = O. We set P = t(I 2n + j) and Q = t(I 2n j), where j [ -oIn °I n For a j-expansive matrix T, of order 2n, the following matrix is defi ned S Since (QT+P) (PT+Q)-IJ ,where p = [O In) J. • P In O 1 S*S = J (QT+P)*-I(T*jT-j)(QT+P)-IJ , P P (1; 1) we can afirm that the matrix T is j-expansive if and only ií the matrix S, defined by (1;1), is contractive. The ...
MODEL THEORY FOR -CONTRACTIONS, 6 2
Agler's abstract model theory is applied to Cρ, the family of operators with unitary ρ-dilations, where ρ is a fixed number in (0, 2]. The extremals, which are the collection of operators in Cρ with the property that the only extensions of them which remain in the family are direct sums, are characterized in a variety of manners. They form a part of any model, and in particular, of the boundary, which is defined as the smallest model for the family. Any model for a family is required to be closed under direct sums, restrictions to reducing subspaces, and unital * -representations. In the case of the family Cρ with ρ ∈ (0, 1) ∪ (1, 2], this closure is shown to be all of Cρ.