Riassunto
The present lectures are based on a course deli vered by the authors at the Uni versi ty of Bucharest, in the winter semester 1985-1986. Without aiming at completeness, the topics selected cover all the major questions concerning hyponormal operators. Our main purpose is to provide the reader with a straightforward access to an active field of research which is strongly related to the spectral and perturbation theories of Hilbert space operators, singular integral equations and scattering theory. We have in view an audience composed especially of experts in operator theory or integral equations, mathematical physicists and graduate students. The book is intended as a reference for the basic results on hyponormal operators, but has the structure of a textbook. Parts of it can also be used as a second year graduate course. As prerequisites the reader is supposed to be acquainted with the basic principles of functional analysis and operator theory as covered for instance by Reed and Simon [1]. A t several stages of preparation of the manuscript we were pleased to benefit from proper comments made by our cOlleagues: Grigore Arsene, Tiberiu Constantinescu, Raul Curto, Jan Janas, Bebe Prunaru, Florin Radulescu, Khrysztof Rudol, Konrad Schmudgen, Florian-Horia Vasilescu. We warmly thank them all. We are indebted to Professor Israel Gohberg, the editor of this series, for his constant encouragement and his valuable mathematical advice. We wish to thank Mr. Benno Zimmermann, the Mathematics Editor at Birkhauser Verlag, for cooperation and assistance during the preparation of the manuscript.
Contenuti
I: Subnormal operators.- 1. Elementary properties and examples.- 2. Characterization of subnormality.- 3. The minimal normal extension.- 4. Putnam’s inequality.- 5. Supplement: Positive definite kernels.- Notes.- Exercises.- II: Hyponormal operators and related objects.- 1. Pure hyponormal operators.- 2. Examples of hyponormal operators.- 3. Contractions associated to hyponormal operators.- 4. Unitary invariants.- Notes.- Exercises.- III: Spectrum, resolvent and analytic functional calculus.- 1. The spectrum.- 2. Estimates of the resolvent function.- 3. A sharpened analytic functional calculus.- 4. Generalized scalar extensions.- 5. Local spectral properties.- 6. Supplement: Pseudo-analytic extensions of smooth functions.- Notes.- Exercises.- IV: Some invariant subspaces for hyponormal operators.- 1. Preliminaries.- 2. Scott Brown’s theorem.- 3. Hyperinvariant subspaces for subnormal operators.- 4. The lattice of invariant subspaces.- Notes.- Exercises.- V: Operations with hyponormal operators.- 1. Operations.- 2. Spectral mapping results.- Notes.- Exercises.- VI: The basic inequalities.- 1. Berger and Shaw’s inequality.- 2. Putnam’s inequality.- 3. Commutators and absolute continuity of self-adjoint operators.- 4. Kato’s inequality.- 5. Supplement: The structure of absolutely continuous self-adjoint operators.- Notes.- Exercises.- VII: Functional models.- 1. The Hilbert transform of vector valued functions.- 2. The singular integral model.- 3. The two-dimensional singular integral model.- 4. The Toeplitz model.- 5. Supplement: One dimensional singular integral operators.- a) The Cauchy integral.- b) The Sohotskii — Plemelj — Privalov formulae.- c) The Hilbert transform on L2(R).- d) Singular integral operators with continuous symbol.- e) The Riemann — Hilbert problem.- Notes.- Exercises.- VIII: Methods of perturbation theory.- 1. The phase shift.- 2. Abstract symbol and Friedrichs operations.- 3. The Birman — Kato — Rosenblum scattering theory.- 4. Boundary behaviour of compressed resolvents.- 5. Supplement: Integral representations for a class of analytic functions defined in the upper half-plane.- Notes.- Exercises.- IX: Mosaics.- 1. The phase operator.- 2. Determining functions.- 3. The principal function.- 4. Symbol homomorphisms and mosaics.- 5. Properties of the mosaic.- 6. Supplement: A spectral mapping theorem for the numerical range.- Notes.- Exercises.- X: The principal function.- 1. Bilinear forms with the collapsing property.- 2. Smooth functional calculus modulo trace-class operators and the trace formula.- 3. The properties of the principal function.- 4. Berger’s estimates.- Notes.- Exercises.- XI: Operators with one dimensional self-commutator.- 1. The global local resolvent.- 2. The kernel function.- 3. A functional model.- 4. The spectrum and the principal function.- Notes.- Exercises.- XII: Applications.- 1. Pairs of unbounded self-adjoint operators.- 2. The Szego limit theorem.- 3. A two dimensional moment problem.- Notes.- Exercises.- References.- Notation and symbols.
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