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. Shmulʹyan 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].