The Gohberg Anniversary Collection Volume II: Topics in Analysis and Operator Theory

In this article we shall use two special classes of reproducing kernel Hilbert spaces (which originate in the work of de Branges [dB) and de Branges-Rovnyak [dBRl), respectively) to solve matrix versions of a number of classical interpolation problems. Enroute we shall reinterpret de Branges' characterization of the first of these spaces, when it is finite dimensional, in terms of matrix equations of the Liapunov and Stein type and shall subsequently draw some general conclusions on rational m x m matrix valued functions which are "J unitary" a.e. on either the circle or the line. We shall also make some connections with the notation of displacement rank which has been introduced and extensively studied by Kailath and a number of his colleagues as well as the one used by Heinig and Rost [HR). The first of the two classes of spaces alluded to above is distinguished by a reproducing kernel of the special form K (>.) = J - U(>')JU(w)* (Ll) w Pw(>') , in which J is a constant m x m signature matrix and U is an m x m J inner matrix valued function over ~+, where ~+ is equal to either the open unit disc ID or the open upper half plane (1)+ and Pw(>') is defined in the table below.

Table of contents of Volume II.- Duality methods for the boundary control of some evolution equations.- Unitary extensions of isometries and contractive intertwining dilations.- Factorization and general properties of nonlinear Toeplitz operators.- Quasilocal algebras over index sets with a minimal condition.- The analogue of Kuroda’s theorem for n-tuples.- The geometry of representing measures and their critical values.- Boundary element analysis of a direct method for the biharmonic Dirichlet problem.- Nonlinear lifting theorems, integral representations and stationary processes in algebraic scattering systems.- Characteristic functions of unitary colligations and of bounded operators in Krein spaces.- Differential operators of fractional order and boundary value problems in the complex domain.- On reproducing kernel spaces, J unitary matrix functions, interpolation and displacement rank.- On asymptotic Toeplitz and Hankel operators.- Iterative commutant lifting for systems with rational symbol.- On the reduction of coercive singular perturbations to regular perturbations.- Averaging techniques for the transport operator and an existence theorem for the BGK equation in kinetic theory.- Factorization of nonlinear system.- Minimal lower separable representations: characterization and construction.- On the inclination of hyperinvariant subspaces of C11- contractions.- Unimodular Möbius-invariant contractive divisors for the Bergman space.- Trigonometric approximation of solutions of periodic pseudodifferential equations.- Wiener-Hopf factorization of certain non-rational matrix functions in mathematical physics.- Classes of operator monotone functions ans Stieltjes functions.- A unified approach to function models, and the transcription problem.- Quadrature methodsfor strongly elliptic Cauchy singular integral equations on an interval.- General Wiener-Hopf operators and representation of their generalized inverses.- Exposed points in H1, I.- Geometrical properties of a unit sphere of the operator spaces in Lp.- C*-algebras of Crystal groups.- On Wiener-Hopf determinants.- Table of contents of Volume I.- Errata.