Defect Functions of Holomorphic Contractive Operator Functions and the Scattering Suboperator through the Internal Channels of a System. Part I

Abstract

Let θ(ζ ) be a Schur operator function, i.e., it is defined and holomorphic on the unit disk D := {ζ ∈ C : |ζ | < 1} and its values are contractive operators acting from one Hilbert space into another one. In the first part of the paper the outer and * -outer Schur operator functions ϕ(ζ ) and ψ(ζ ) which describe respectively the deviations of the function θ(ζ ) from inner and * -inner operator functions are studied. If ϕ(ζ ) = 0, then it means that in the scattering system for which θ(ζ ) is the transfer function a portion of "information" comes inward the system and does not go outward, i.e., it is left in the internal channels of the system (Sect. 6). The function ψ(ζ ) has the analogous property for the dual system. For this reason these functions are called the defect functions of the function θ(ζ ). The explicit form of the defect functions ϕ(ζ ) and ψ(ζ ) is obtained and the analytic connection of these functions with the function θ(ζ ) is described (Sects. 3, 5). The operator functions ϕ(ζ ) θ(ζ ) and(ψ(ζ ), θ (ζ )) are Schur functions as well (Sect. 3). It is important that there exists the unique contractive measurable operator function χ(t), t ∈ ∂D, such that the operator function, t ∈ ∂D, is also contractive (Part II, Sect. 12). The second part of the paper is devoted to studying the properties of the function χ(t). Specifically, it is shown that the function χ(t) is the scattering suboperator through the internal channels of the scattering system for which θ(ζ ) is the transfer function (Part II, Sect. 12).

Minimal Passive Realizations of Generalized Schur Functions in Pontryagin Spaces

“…on T. The function ϕ S is called the right defect function of S. The function ψ S is called the left defect function of S. Both ϕ S and ψ S are unique up to a unitary constant. The theory of the defect functions is considered, for instance, in [16,17,18]. Various connections of defect functions and passive realizations can be found in [4,9,10].…”

“…In what follows, the range of ϕ θ and the domain of ψ θ will be denoted, respectively, by D ϕ θ and D ψ θ . In the Hilbert space settings, it is well known [17,18] that for a standard Schur function θ ∈ S(U, Y), there exists a function χ θ ∈ L ∞ (D ψ θ , D ϕ θ ) such that the function Θ(ζ) := θ(ζ) ψ θ (ζ) ϕ θ (ζ) χ θ (ζ) (4.21)…”

“…ζ on the unit circle T, it holds ϕ * (ζ)ϕ(ζ) ≤ I − S * (ζ)S(ζ), and for every other operator valued analytic function ϕ with similar property, it holds ϕ * (ζ) ϕ(ζ) ≤ ϕ * (ζ)ϕ(ζ). For the existence of defect functions, see [30,Theorem V.4.2], and for a detailed treatise, see [16,17,18]. In [7], Arov et al constructed ( * -)optimal minimal passive systems in the Hilbert space setting without using defect functions.…”