Ebook: Theory of Commuting Nonselfadjoint Operators

Considering integral transformations of Volterra type, F. Riesz and B. Sz.-Nagy no­ ticed in 1952 that [49]: "The existence of such a variety of linear transformations, having the same spectrum concentrated at a single point, brings out the difficulties of characterization of linear transformations of general type by means of their spectra." Subsequently, spectral analysis has been developed for different classes of non­ selfadjoint operators [6,7,14,20,21,36,44,46,54]. It was then realized that this analysis forms a natural basis for the theory of systems interacting with the environment. The success of this theory in the single operator case inspired attempts to create a general theory in the much more complicated case of several commuting operators with finite-dimensional imaginary parts. During the past 10-15 years such a theory has been developed, yielding fruitful connections with algebraic geometry and sys­ tem theory. Our purpose in this book is to formulate the basic problems appearing in this theory and to present its main results. It is worth noting that, in addition to the joint spectrum, the corresponding algebraic variety and its global topological characteristics play an important role in the classification of commuting operators. For the case of a pair of operators these are: 1. The corresponding algebraic curve, and especially its genus. 2. Certain classes of divisors - or certain line bundles - on this curve.

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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.

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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.

Content:
Front Matter....Pages i-xvii
Front Matter....Pages 1-1
Preliminary Results....Pages 3-17
Colligations and Vessels....Pages 18-28
Open Systems and Open Fields....Pages 29-41
The Generalized Cayley — Hamilton Theorem....Pages 42-70
Front Matter....Pages 71-71
Joint Spectrum and the Spectral Mapping Theorem....Pages 73-80
Joint Spectrum of Commuting Operators with Compact Imaginary Parts....Pages 81-91
Properties of Discriminant Varieties of a Commutative Vessel....Pages 92-100
Front Matter....Pages 101-101
Operator Colligations and Vessels in Banach Space....Pages 103-130
Bezoutian Vessels in Banach Space....Pages 131-184
Front Matter....Pages 185-185
Characteristic Functions of Two-Operator Vessels in a Hilbert Space....Pages 187-233
The Determinantal Representations and the Joint Characteristic Functions in the Case of Real Smooth Cubics....Pages 234-255
Triangular Models for Commutative Two-Operator Vessels on Real Smooth Cubics....Pages 256-301
Back Matter....Pages 303-318

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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.

Content:
Front Matter....Pages i-xvii
Front Matter....Pages 1-1
Preliminary Results....Pages 3-17
Colligations and Vessels....Pages 18-28
Open Systems and Open Fields....Pages 29-41
The Generalized Cayley — Hamilton Theorem....Pages 42-70
Front Matter....Pages 71-71
Joint Spectrum and the Spectral Mapping Theorem....Pages 73-80
Joint Spectrum of Commuting Operators with Compact Imaginary Parts....Pages 81-91
Properties of Discriminant Varieties of a Commutative Vessel....Pages 92-100
Front Matter....Pages 101-101
Operator Colligations and Vessels in Banach Space....Pages 103-130
Bezoutian Vessels in Banach Space....Pages 131-184
Front Matter....Pages 185-185
Characteristic Functions of Two-Operator Vessels in a Hilbert Space....Pages 187-233
The Determinantal Representations and the Joint Characteristic Functions in the Case of Real Smooth Cubics....Pages 234-255
Triangular Models for Commutative Two-Operator Vessels on Real Smooth Cubics....Pages 256-301
Back Matter....Pages 303-318
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