Recent Advances in Operator Theory and Applications

the recent IWOTA meetings, IWOTA 2006 was focused on a few special themes, without loss of the general IWOTA mission. Our special interest areas were Hilbert/Krein space operator theory; Complex function theory related to Hilbert space operators; Systems theory related to Hilbert space operators. This volume contains 16 contributions, which re? ect the recent development in operator theory and applications. The organizers gratefully acknowledge the support of the following institutions: KRF (Korea Research Foundation); Department of Mathematics, Seoul National University; Research Institute of Mathematics, Seoul National University. Tsuyoshi Ando, Raul Curto Il Bong Jung, Woo Young Lee (Editors) OperatorTheory: Advances andApplications, Vol. 187, 1-16 c 2008Birkh auserVerlagBasel/Switzerland AConnectionbetweenSzeg oandNehari SequencesintheMatrix-valuedCase Daniel Alpay and Israel Gohberg Abstract. One can associate to a rational function which is moreover strictly positive on the unit circle two sequences of numbers in the open unit disk, called the Szeg o sequence and the Nehari sequence. In the scalar case, they coincide up to multiplication by? 1. We study the corresponding result in the matrix-valued case. Mathematics Subject Classi? cation (2000). Primary: 34A55, 49N45, 70G30; Secondary: 93B15, 47B35. Keywords. Inverse problems, scattering matrix, Schurparameters, state space method, extension problems. 1. Introduction Letw(z) be a scalar rational function strictly positive on the unit circle. One can associate to it an in? nite sequence of numbers in the open unit disk, called in [1] a Szeg o sequence. This sequence characterizes in a unique wayw(z)providedsome normalization is chosen; we will take 2? 1 it w(e )dt=1.

A Connection between Szegö and Nehari Sequences in the Matrix-valued Case. - Multivariable Generalizations of the Schur Class: Positive Kernel Characterization and Transfer Function Realization. - Some Problems Concerning the Test Functions in the Szegö and Avram-Parter Theorems. - On the Numerical Solution of a Hypersingular Integral Equation with Fixed Singularities. - Spectrum and Principal Function of Operators. - Spectral Consequences due to General Hölder s Inequalities and Domination of Semigroups. - Quadratically Hyponormal Recursively Generated Weighted Shifts. - Relaxed Commutant Lifting: An Equivalent Version and a New Application. - Variants of Ando-Hiai Inequality. - On Local Spectral Theory. - The Finite Sections Approach to the Index Formula for Band-dominated Operators. - A New Proof that Rudin s Module is not Finitely Generated. - Quasinilpotent Part of class A or (p, k)-quasihyponormal Operators. - A New Majorization Induced by Matrix Order. - H ? Functional Calculus and Characterization of Domains of Fractional Powers. - Operators on Locally Convex Spaces.