Shift operators contained in contractions and pseudocontinuable matrixvalued Schur functions

Abstract

We show that the pseudocontinuability of a matrix-valued Schur function θ in the unit disk is completely determined by the properties of the maximal shift and maximal coshift contained in a corresponding contractive operator T which has the characteristic operator function θ. ). In this way, there arises the natural question: how is the pseudocontinuability of the Schur function θ related to the properties of the fundamental operator T of the corresponding scattering system? The main goal of this paper is to study these interrelations. Hereby we will concentrate on the discussion of matrix-valued Schur functions. The main results of this paper are Theorem 3.7 and Theorem 3.9 and some consequences following from them. These theorems show that the existence of a pseudocontinuation of the Schur function θ is completely determined by the properties of the maximal shift and maximal coshift contained in the fundamental operator T of the scattering system. Namely, this interrelation is what we expressed in the title of the paper.Our approach is based on a fundamental result due to Arov [1] and, independently, Douglas and Helton [13]. Their theorem says that a Schur function θ is pseudocontinuable if and only if it can be represented as block of a two-sided inner operator-valued function A (see Theorem 3.1). It turns out (see Theorem 3.20) that there is a bijective correspondence between the minimal two-sided inner extensions A of θ and the orthogonal shifts and coshifts contained in T.It should be mentioned that the shifts and coshifts contained in T can be interpreted as inner channels of the corresponding scattering system. Our technique of constructing two-sided inner extensions A of θ is based on a transition from a system with loss to some conservative system by transforming inner channels into outer ones. *

Shift operators contained in contractions, pseudocontinuable Schur functions and orthogonal systems on the unit circle

“…The present paper continues recent investigations on several questions of pseudocontinuability of Schur functions in the unit disk (see [3] and [7]). Our main aim is to establish a direct relation between two well-known criteria for pseudocontinuability of non-inner Schur functions in the unit disk (see Theorem 3.9 and Theorem 4.2).…”

The S-recurrence of Schur Parameters of Non-inner Rational Schur Functions

“…Note that this approach is based on the description in terms of the Schur parameters of the relative position of the largest shift and the largest coshift in a completely nonunitary contraction (see [19,Section 3]). It should be mentioned that these results received a further development in [8,[21][22][23][24].…”

“…This approach is based on the description in terms of the Schur parameters of the relative position of the largest shift and the largest coshift in a completely nonunitary contraction. It should be mentioned that these results received a further development in [8,[21][22][23][24].This paper is aimed to give a survey about essential results on this direction. The main object in the approach is based on considering a Schur function as characteristic function of a contraction (see Section 1.2).…”

Defect Functions of Holomorphic Contractive Operator Functions and the Scattering Suboperator through the Internal Channels of a System. Part I

“…coshift V T ) has the multiplicity zero. Theorem 1.9 [9,13] Let T be a completely nonunitary contraction on H. Then the multiplicities of the maximal shifts V T and V T * are not greater than δ T * and δ T , respectively. The next statements are very important (see, e.g., [11]): …”