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 contractive-operator 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 Schur-class functions which arise as the scattering matrix for a certain type of (not necessarily minimal) Lax--Phillips multievolution scattering system having some additional geometric 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.