Lectures on Hyponormal Operators

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.