Generalized fractional-linear transformations of operator balls

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

“…Example 3.5 below presents another example where R Φ does not preserve ∞ ≺ with Φ an operator polynomial of degree one. Note that one can view R Φ in the context of the Redheffer maps R M of [17] by taking M = T Φ , i.e., M ij = T Φij , and A = T F . However, the condition Φ 11 (0)F (0) < 1 does not imply that I − T Φ11 T F is (boundedly) invertible, and hence R Φ acts on a larger domain than R TΦ .…”

“…In Section 1 we discuss various definitions and implications of the pre-order ≺ and equivalence relation ∼ of [17] and derive relations between the parameters in the different definitions. The specification of ≺ and ∼ to contractive Toeplitz and Laurent operators, both analytic and nonanalytic, leading to the definition of the pre-order We conclude this introduction with some words on notation and terminology.…”

“…In [17] Yu.L. Shmul'yan considered the pre-order relation on the set of contractions L 1 (H 1 , H 2 ) defined in the following theorem.…”

“…In a 1980 paper [17] Yu.L. Shmul'yan introduced a pre-order relation on the set of Hilbert space contractions, and showed that linear fractional maps of Redheffer type, as initially studied by R.M.…”

“…With L 1 (H 1 , H 2 ) we denote the set of contractions from H 1 to H 2 , and, given C ∈ L 1 (H 1 , H 2 ), we write D C for the defect operator D C = (I − C * C) [17] this is denoted by B ≺ A, however, more in line with other pre-orders on L 1 (H 1 , H 2 ), the order is reversed in [11], and we will adopt the notation of [11] here.) Section 1 contains a detailed discussion of the pre-order ≺ and the corresponding equivalence relation ∼.…”