I: Subnormal operators -- 1. Elementary properties and examples -- 2. Characterization of subnormality -- 3. The minimal normal extension -- 4. Putnamโs 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 Brownโs 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 Shawโs inequality -- 2. Putnamโs inequality -- 3. Commutators and absolute continuity of self-adjoint operators -- 4. Katoโs 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 -- Notes -- Exercises -- VIII: Methods of perturbation theory -- 1. The phase shift -- 2. Abstract symbol and Friedrichs operations -- 3. The Birman โ Kato โ 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. Bergerโs 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

1. Mathematics
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