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A pre-order and equivalence relation on the class of positive real Hilbertspace operators are introduced, in correspondence with similar relations forcontraction operators defined by Yu.L. Shmul'yan in [7]. It is shown that thepre-order, and hence the equivalence relation, are preserved by certain linearfractional transformations. As an application, the operator relations areextended 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 theassociated linear fractional transformations of $\fC(\cU)$ preserve theequivalence relation on their natural domain of definition, but not necessarilythe pre-order, paralleling similar results for Schur class functions in [3].

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