Mihai Putinar Bücher

This book exploits the classification of a class of linear bounded operators with rank-one self-commutators in terms of their spectral parameter, known as the principal function. The resulting dictionary between two dimensional planar shapes with a degree of shade and Hilbert space operators turns out to be illuminating and beneficial for both sides. An exponential transform, essentially a Riesz potential at critical exponent, is at the heart of this novel framework; its best rational approximants unveil a new class of complex orthogonal polynomials whose asymptotic distribution of zeros is thoroughly studied in the text. Connections with areas of potential theory, approximation theory in the complex domain and fluid mechanics are established. The text is addressed, with specific aims, at experts and beginners in a wide range of areas of current interest: potential theory, numerical linear algebra, operator theory, inverse problems, image and signal processing, approximation theory, mathematical physics.

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