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Mathematics and Its Applications #332: Theory of Commuting Nonselfadjoint Operatorsby M. S. Livsic
Synopses & ReviewsPublisher Comments:Theory of Commuting Nonselfadjoint Operators presents a systematic and cogent exposition of results hitherto only available as research articles. The recently developed theory has revealed important and fruitful connections with the theory of collective motions of systems distributed continuously in space and with the theory of algebraic curves.
A rigorous mathematical definition of the physical concept of a particle is proposed, and a concrete image of a particle conceived as a localised entity in space is obtained. The duality of waves and particles then becomes a simple consequence of general equations of collective motions: particles are collective manifestations of inner states; waves are guiding waves of particles. The connection with the theory of algebraic curves is also important. For wide classes of pairs of commuting nonselfadjoint operators there exists the notion of a `discriminant' polynomial of two variables which generalises the classical notion of the characteristic polynomial for a single operator. A given pair of operators annihilate their discriminant. Divisors of corresponding line bundles play the main role in the classification of commuting operators. Audience: Researchers and postgraduate students in operator theory, system theory, quantum physics and algebraic geometry. Book News Annotation:During the past 1015 years, a theory of commuting nonselfadjoint operators has been developed, which has yielded important connections with both the theory of collective motions of systems distributed continuously in space and with the theory of algebraic curves. This volume formulates the basic problems appearing in this theory and presents its main results. The volume contains 12 chapters arranged in four parts: operator vessels in Hilbert space; joint spectrum and discriminant varieties of a commutative vessel; operator vessels in Banach space; and spectral analysis of twooperator vessels.
Annotation c. Book News, Inc., Portland, OR (booknews.com) Synopsis:This volume presents a systematic exposition of results hitherto only available as research articles. It is intended for researchers and postgraduate students in operator theory, system theory, quantum physics and algebraic geometry.
Synopsis:Theory of Commuting Nonselfadjoint Operators presents a systematic and cogent exposition of results hitherto only available as research articles. The recently developed theory has revealed important and fruitful connections with the theory of collective motions of systems distributed continuously in space and with the theory of algebraic curves. A rigorous mathematical definition of the physical concept of a particle is proposed, and a concrete image of a particle conceived as a localised entity in space is obtained. The duality of waves and particles then becomes a simple consequence of general equations of collective motions: particles are collective manifestations of inner states; waves are guiding waves of particles. The connection with the theory of algebraic curves is also important. For wide classes of pairs of commuting nonselfadjoint operators there exists the notion of a `discriminant' polynomial of two variables which generalises the classical notion of the characteristic polynomial for a single operator. A given pair of operators annihilate their discriminant. Divisors of corresponding line bundles play the main role in the classification of commuting operators. Audience: Researchers and postgraduate students in operator theory, system theory, quantum physics and algebraic geometry.
Table of ContentsPreface. Introduction. I: Operator Vessels in Hilbert Space. 1. Preliminary results. 2. Colligations and vessels. 3. Open systems and open fields. 4. The generalized CayleyHamilton theorem. II: Joint Spectrum and Discriminant Varieties of a Commutative Vessel. 5. Joint spectrum and the spectral mapping theorem. 6. Joint spectrum of commuting operators with compact imaginary parts. 7. Properties of discriminant varieties of a commutative vessel. III: Operator Vessels in Banach Spaces. 8. Operator colligations and vessels in Banach space. 9. Bezoutian vessels in Banach space. IV: Spectral Analysis of TwoOperator Vessels. 10. Characteristic functions of twooperator vessels in a Hilbert space. 11. The determinantal representations and the joint characteristic functions in the case of real smooth cubics. 12. Triangular models for commutative twooperator vessels on real smooth cubics. References. Index.
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