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

A pre-order and equivalence relation on the class of Hilbert space operators with positive real part are introduced, in correspondence with similar relations for contraction operators defined by Yu.L. Shmul'yan in [6]. It is shown that the pre-order, and hence the equivalence relation, is preserved by certain linear fractional transformations. As an application, the operator relations are extended to the class C(U) of Carathéodory functions on the unit disc D of C whose values are operators on a finite dimensional Hilbert space U. With respect to these relations on C(U) it turns out that the associated linear fractional transformations of C(U) 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].