Lectures on Hyponormal Operators

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The book covers a comprehensive range of topics related to subnormal and hyponormal operators, structured into twelve distinct sections. It begins with foundational concepts, including elementary properties, characterizations of subnormality, and the minimal normal extension, alongside Putnam’s inequality and positive definite kernels. The exploration of hyponormal operators follows, detailing pure hyponormal operators, their examples, and associated contractions. The spectrum, resolvent, and analytic functional calculus are examined, focusing on spectrum characterization, resolvent estimates, and generalized scalar extensions. Invariant subspaces for hyponormal operators are discussed, including Scott Brown’s theorem and hyperinvariant subspaces. The text also addresses operations with hyponormal operators and presents key spectral mapping results. Basic inequalities relevant to the study are outlined, including Berger and Shaw’s inequality, commutators, and Kato’s inequality. Functional models are introduced, featuring the Hilbert transform and various singular integral models. Perturbation theory methods are explored, including phase shifts and scattering theory. The concept of mosaics is presented, detailing phase operators and determining functions. The principal function is analyzed through bilinear forms and smooth functional calculus. Lastly, applications to unbounded self-adjoint operators and moment problems are d