A pre-order and an equivalence relation on Schur class functions and their invariance under linear fractional transformations

TL;DR

This paper introduces a pre-order and an equivalence relation on operator-valued Schur class functions, examining how Redheffer linear fractional transformations affect these relations, highlighting invariance and conditions for preservation.

Contribution

It defines new relations on Schur class functions and analyzes their invariance properties under Redheffer LFTs, extending previous work by Yu.L. Shmul'yan.

Findings

Redheffer LFTs preserve the equivalence relation.

The pre-order is preserved under additional conditions.

The study deepens understanding of transformations on Schur functions.

Abstract

Motivated by work of Yu.L. Shmul'yan a pre-order and an equivalence relation on the set of operator-valued Schur class functions are introduced and the behavior of Redheffer linear fractional transformations (LFTs) with respect to these relations is studied. In particular, it is shown that Redheffer LFTs preserve the equivalence relation, but not necessarily the pre-order. The latter does occur under some additional assumptions on the coefficients in the Redheffer LFT.