The matricial Schur problem in both nondegenerate and degenerate cases

Abstract

Key wordsThe principal object of this paper is to present a new approach simultaneously to both nondegenerate and degenerate cases of the matricial Schur problem. This approach is based on an analysis of the central matrixvalued Schur functions which was started in [24]-[26] and then continued in [27]. In the nondegenerate situation we will see that the parametrization of the solution set obtained here coincides with the well-known formula of D. Z. Arov and M. G. Kreȋn for that case (see [1]). Furthermore, we give some characterizations of the situation that the matricial Schur problem has a unique solution. The main goal of this paper is to present an approach to the matricial version of the classical Schur problem (see [32] and [33]) in both nondegenerate and degenerate cases. Roughly speaking, our strategy is due to an adaptation of the authors' former investigations to the general (i.e. possibly degenerate) matricial Carathéodory problem (see [29]). Our method is essentially based on the first and second authors former investigations [23]-[27] on the central matrix-valued Schur function associated with a finite Schur sequence of complex matrices. In particular, we will make frequently use of the matrix ball description of the elements of matricial Schur sequences. The main results of this paper (see Theorems 1.1, 3.2, 3.6, and 4.2) contain descriptions of the solution set of a matricial Schur problem in terms of a linear fractional transformation, the generating matrix-valued function of which is a matrix polynomial. The canonical blocks of this matrix polynomial will be constructed with the aid of those quadruple of matrix polynomials which were used in [27] to derive right and left quotient representations of central matrix-valued Schur functions (see Theorem 2.4).A different approach to the degenerate matricial Schur problem was used in [11]-[16] of V. K. Dubovoj. His method is based on a combination of V. P. Potapov's fundamental matrix inequality approach with the technique of subspaces of type K which was created by V. K. Dubovoj to handle the specific of the matricial Schur problem *

On the Weyl Matrix Balls Associated with the Matricial Schur Problem in the Nondegenerate and Degenerate Cases

“…This paper is closely related to the investigations in [9] on the matricial Schur problem in both nondegenerate and degenerate cases. In [9] a parametrization of the solution set of a general (possibly degenerate) matricial Schur problem was obtained in terms of a linear fractional transformation.…”

“…Since we see from Proposition 3.3 that the matrices −wθ * n (w) and −wθ * n (w) are both contractive, [9,Proposition 3.4] provides us…”

“…In [9] a parametrization of the solution set of a general (possibly degenerate) matricial Schur problem was obtained in terms of a linear fractional transformation. Using this parametrization we study the set of values attained by the solutions of a matricial Schur problem in a prescribed fixed point of the open unit disk.…”

“…The case of a degenerate Schur problem was first treated by S. N. Zinenko [22] who used essentially the description of the solution set which was constructed in [3, part IV]. Our computation of the corresponding Weyl matrix balls is based on the approach to the matricial Schur problem developed in [9]. The key instrument is here a distinguished rational matrix function θn which is associated with an arbitrarily given matricial Schur problem.…”

The Schur Problem and Block Operator CMV Matrices

“…, is the central solutions to the Schur problems [21], [25], [7]. Parametrization (5.1) is similar to known parameterizations [11], [14], [22], [23], [24], [25], [26] which are obtained by another methods.…”