Theory of Commuting Nonselfadjoint Operators by M.S. LivsicTheory of Commuting Nonselfadjoint Operators by M.S. Livsic

Theory of Commuting Nonselfadjoint Operators

byM.S. Livsic, N. Kravitsky, A.S. Markus

Paperback | December 9, 2010

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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.
Title:Theory of Commuting Nonselfadjoint OperatorsFormat:PaperbackDimensions:318 pages, 23.5 × 15.5 × 0.01 inPublished:December 9, 2010Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9048145856

ISBN - 13:9789048145850


Table of Contents

Preface. Introduction. I: Operator Vessels in Hilbert Space. 1. Preliminary results. 2. Colligations and vessels. 3. Open systems and open fields. 4. The generalized Cayley-Hamilton 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 Two-Operator Vessels. 10. Characteristic functions of two-operator 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 two-operator vessels on real smooth cubics. References. Index.