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

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 $\text{fC}(\text{cU})$ of Carath'eodory functions on the unit disc $\text{BD}$ of $\text{BC}$ whose values are operators on a finite dimensional Hilbert space $\text{cU}$. With respect to these relations on $\text{fC}(\text{cU})$ it turns out that the associated linear fractional transformations of $\text{fC}(\text{cU})$ 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].