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Motivated by work of Yu.L. Shmul'yan a pre-order and an equivalence relationon the set of operator-valued Schur class functions are introduced and thebehavior of Redheffer linear fractional transformations (LFTs) with respect tothese relations is studied. In particular, it is shown that Redheffer LFTspreserve the equivalence relation, but not necessarily the pre-order. Thelatter does occur under some additional assumptions on the coefficients in theRedheffer LFT.

A pre-order and an equivalence relation on Schur class functions and their invariance under linear fractional transformations