Lectures on Hyponormal Operators

InhaltsverzeichnisI: 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.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.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.V: Operations with hyponormal operators.1. Operations.2. Spectral mapping results.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.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.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.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.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.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.XII: Applications.1. Pairs of unbounded self-adjoint operators.2. The Szego limit theorem.3. A two dimensional moment problem.References.Notation and symbols.