A pre-order on positive real operators and its invariance under linear fractional transformations

TL;DR

This paper introduces a new pre-order and equivalence relation for positive real Hilbert space operators, demonstrating their invariance under certain linear fractional transformations and extending these concepts to Carathéodory functions.

Contribution

It defines a novel pre-order and equivalence relation for positive real operators and shows their invariance under specific transformations, extending the theory to operator-valued Carathéodory functions.

Findings

Pre-order and equivalence relation are preserved by certain linear fractional transformations.

Relations extend to Carathéodory functions on the unit disc.

Equivalence relation is preserved, but the pre-order may not be, for these functions.

Abstract

A pre-order and equivalence relation on the class of positive real Hilbert space operators are introduced, in correspondence with similar relations for contraction operators defined by Yu.L. Shmul'yan in [7]. It is shown that the pre-order, and hence the equivalence relation, are preserved by certain linear fractional transformations. As an application, the operator relations are extended to the class $\fC(\cU)$ of Carath\'eodory functions on the unit disc $\BD$ of $\BC$ whose values are operators on a finite dimensional Hilbert space $\cU$. With respect to these relations on $\fC(\cU)$ it turns out that the associated linear fractional transformations of $\fC(\cU)$ preserve the equivalence relation on their natural domain of definition, but not necessarily the pre-order, paralleling similar results for Schur class functions in [3].