This book is devoted to some topical problems and applications of operator theory and its interplay with modern complex analysis. It consists of 20 selected survey papers that represent updated (mainly plenary) addresses to the IWOTA 2000 conference held at Bordeaux from June 13 to 16, 2000. The main subjects of the volume include: - spectral analysis of periodic differential operators and delay equations, stabilizing controllers, Fourier multipliers; - multivariable operator theory, model theory, commutant lifting theorems, coisometric realizations; - Hankel operators and forms; - operator algebras; - the Bellman function approach in singular integrals and harmonic analysis, singular integral operators and integral representations; - approximation in holomorphic spaces. These subjects are unified by the common "operator theoretic approach" and the systematic use of modern function theory techniques.
Table of ContentsThe Schur algorithm for generalized Schur functions I: coisometric realizations.- Hardy spaces on a finite bordered Riemann surface, multivariable operator model theory and Fourier analysis along a unimodular curve.- On Hankel operators associated with Markov functions.- Threshold effects near the lower edge of the spectrum for periodic differential operators of mathematical physics.- Cauchy’s singular integral operator and its beautiful spectrum.- Approximation problems in some holomorphic spaces, with applications.- The Bargmann isometry and GaborDaubechies wavelet localization operators.- Stabilization of irrational transfer functions by controllers with internal loop.- Free Semigroup Algebras. A Survey.- Some geometric invariants from resolutions of Hilbert modules.- Some remarks on spherical isometries.- Higher-order Hankel forms and cohomology groups.- A weighted version of almost commutant lifting.- Hyperbolicity of semigroups and Fourier multipliers.- Relaxations of quadratic programs in operator theory and system analysis.- Bellman function in stochastic control and harmonic analysis.- Integral representations and uniqueness sets for star-invariant subspaces.- Well-posed linear systems, LaxPhillips scattering, and Lp-multipliers.- Spectral theory for delay equations.- On a singular perturbation of a multiplication operator.