Shift Operators Contained in Contractions, Pseudocontinuable Schur Functions and Orthogonal Systems on the Unit Circle

TL;DR

This paper explores the relationship between criteria for pseudocontinuability of Schur functions and their associated orthogonal systems on the unit circle, linking function theory with operator models and orthogonal basis constructions.

Contribution

It establishes a direct connection between two known criteria for pseudocontinuability of Schur functions via orthogonal basis representations and operator-theoretic perspectives.

Findings

Reformulation of criteria in terms of orthogonal basis

Construction of a special orthogonal basis related to the measure mu

Clarification of the connection between criteria and operator models

Abstract

The main aim of this paper is to establish the connection between well-known criteria for the pseudocontinuability of a non-inner Schur function Theta in the unit disk. In a canonical way we associate a probability measure mu on the unit circle with Theta. One of the two criteria will be reformulated in the face of mu, whereas the other one is drafted in view of a completely non--unitary contraction T having Theta as its corresponding characteristic function. Our main result clarifies an immediate connection between the above-mentioned two criteria. For this reason, we construct a special orthogonal basis in the space of square-integrable functions with respect to mu and rewrite these criteria in terms of this orthogonal basis.