Lectures on Hyponormal Operators

Inhaltsverzeichnis

• 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.
• 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.