Preorders and equivalences on the operator sphere

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

“…The characterizations of ∼ given in the next result again go back to [11], the inequalities for X and Y are new. …”

“…In a later paper by Khatskevich, Shmul'yan and Shul'man [11] it was shown that the formulation of ≺ in (POi) is equivalent to those in (POii) and (POiii), by extending ideas from real convex analysis to the complex numbers. The paper [11] also discusses the relation between the pre-order of Theorem 1.1, the Harnack pre-order from [18] and a pre-order defined by Ceausescu in [4].…”

“…The relations between the constants appearing in (POi)-(POiv) from statement (iii) are necessary for our specification to contractive analytic Laurent operators in Section 2 in terms of their symbols, and require a refinement of many of the arguments used in [11].…”

“…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) (H 1 , H 2 ). (Actually,in [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 ∼.…”

Harnack and Shmul’yan pre-order relations for Hilbert space contractions

“…Both pre-order relations have nice geometric and analytic interpretations. Although these two pre-orders have been around since 1970s and 1980s [1,5,[15][16][17]20,24], their structure is to date not completely understood, and in recent years there has been an increase in interest for this topic [2,7,8,10,12,13,18,19,23].…”