Scattering Systems with Several Evolutions and Formal Reproducing Kernel Hilbert Spaces

Abstract

A Schur-class function in d variables is defined to be an analytic contractiveoperator valued function on the unit polydisk. Such a function is said to be in the Schur-Agler class if it is contractive when evaluated on any commutative d-tuple of strict contractions on a Hilbert space. It is known that the Schur-Agler class is a strictly proper subclass of the Schur class if the number of variables d is more than two. The Schur-Agler class is also characterized as those functions arising as the transfer function of a certain type (Givone-Roesser) of conservative multidimensional linear system. Previous work of the authors identified the Schur-Agler class as those Schurclass functions which arise as the scattering matrix for a certain type of (not necessarily minimal) Lax-Phillips multievolution scattering system having some additional geoCommunicated by Irene Sabadini.C. Sadosky passed away in December 2010. We lost a collaborator of many years, and a wonderful friend. Like many others, we miss her. J. A. Ball et al. metric structure. The present paper links this additional geometric scattering structure directly with a known reproducing-kernel characterization of the Schur-Agler class. We use extensively the technique of formal reproducing kernel Hilbert spaces that was previously introduced by the authors and that allows us to manipulate formal power series in several commuting variables and their inverses (e.g., Fourier series of elements of L 2 on a torus) in the same way as one manipulates analytic functions in the usual setting of reproducing kernel Hilbert spaces.

Hardy-space function theory, operator model theory, and dissipative linear systems: the multivariable, free-noncommutative, weighted Bergman-space setting

“…The following result for the special case where K ′ = k nc,Sz ⊗ I U and K = k nc,Sz ⊗ I Y appears in [27,Theorem 3.15] and in [21,Theorem 3.1]. We give a simple direct proof based on an adaptation of the proof of [26,Proposition 2.10] where a more complicated two-sided commutative setting is studied. Proposition 3.2.…”

Canonical Agler decompositions and transfer function realizations

“…For brevity, this paper only outlines the structure of particular scattering systems defined for Φ ∈ S 2 (E, E * ). Many details of these scattering systems also appear in [14] and [13]. For a review of the general theory of one-and multi-evolution scattering systems, see [14].…”

“…This work was continued in [24] where a specific scattering subspace associated to Φ, denoted K Φ , was used to show that canonical decompositions of K Φ yield Agler kernels (K 1 , K 2 ) of Φ. The analysis from [14] was also extended in [13]; here, many results from [14] are illuminated or extended via the theory of formal reproducing kernel Hilbert spaces.…”

Analytic continuation of concrete realizations and the McCarthy Champagne conjecture

“…φ) and concrete Hilbert space geometry to produce and analyze more specific TFRs. They continued this seminal work with Kaliuzhnyi-Verbovetskyi in [15], which includes an exhaustive analysis of TFRs and connections between the geometric scattering structure and associated formal reproducing kernel Hilbert spaces. Ball and Bolotnikov conducted additional insightful work on canonical TFRs in [13,14].…”

“…The more general class of so-called weakly coisometric realizations for d-variable Schur-Agler functions (Schur functions that also satisfy von Neumann's inequality) and their associated A, B, C, D formulas were studied earlier in[14]. Specifically, Definition 3.1 and Theorem 3.4 in[14] also establish the formulas for A and B above and imply that C and D must each satisfy a so-called structured Gleason problem.The concrete function theory interpretations for C and D in Theorem 2.1 are also related to the technical and extensive work in[15]. In particular, in Theorem 5.9, the authors assume that a given d-variable Schur-Agler function ϕ possesses a so-called minimal augmented Agler decomposition and use it to construct a specific unitary realization for ϕ via the theory of scattering systems and formal reproducing kernel Hilbert spaces.…”