Theory of Commuting Nonselfadjoint Operators

1. Front Matter

   Pages i-xvii

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2. operator Vessels in Hilbert Space

   1. Front Matter

      Pages 1-1

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   2. Preliminary Results

      • M. S. Livšic, N. Kravitsky, A. S. Markus, V. Vinnikov

      Pages 3-17

   3. Colligations and Vessels

      • M. S. Livšic, N. Kravitsky, A. S. Markus, V. Vinnikov

      Pages 18-28

   4. Open Systems and Open Fields

      • M. S. Livšic, N. Kravitsky, A. S. Markus, V. Vinnikov

      Pages 29-41

   5. The Generalized Cayley — Hamilton Theorem

      • M. S. Livšic, N. Kravitsky, A. S. Markus, V. Vinnikov

      Pages 42-70

3. Joint Spectrum and Discriminant Varieties of a Commutative Vessel

   1. Front Matter

      Pages 71-71

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   2. Joint Spectrum and the Spectral Mapping Theorem

      • M. S. Livšic, N. Kravitsky, A. S. Markus, V. Vinnikov

      Pages 73-80

   3. Joint Spectrum of Commuting Operators with Compact Imaginary Parts

      • M. S. Livšic, N. Kravitsky, A. S. Markus, V. Vinnikov

      Pages 81-91

   4. Properties of Discriminant Varieties of a Commutative Vessel

      • M. S. Livšic, N. Kravitsky, A. S. Markus, V. Vinnikov

      Pages 92-100

4. Operator Vessels in Banach Spaces

   1. Front Matter

      Pages 101-101

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   2. Operator Colligations and Vessels in Banach Space

      • M. S. Livšic, N. Kravitsky, A. S. Markus, V. Vinnikov

      Pages 103-130

   3. Bezoutian Vessels in Banach Space

      • M. S. Livšic, N. Kravitsky, A. S. Markus, V. Vinnikov

      Pages 131-184

5. Spectral Analysis of Two-Operator Vessels

   1. Front Matter

      Pages 185-185

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   2. Characteristic Functions of Two-Operator Vessels in a Hilbert Space

      • M. S. Livšic, N. Kravitsky, A. S. Markus, V. Vinnikov

      Pages 187-233

   3. The Determinantal Representations and the Joint Characteristic Functions in the Case of Real Smooth Cubics

      • M. S. Livšic, N. Kravitsky, A. S. Markus, V. Vinnikov

      Pages 234-255

   4. Triangular Models for Commutative Two-Operator Vessels on Real Smooth Cubics

      • M. S. Livšic, N. Kravitsky, A. S. Markus, V. Vinnikov

      Pages 256-301

6. Back Matter

   Pages 303-318

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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.

• Banach space
• Divisor
• Grad
• Hilbert space
• Operator theory
• algebraic curve
• algebraic geometry
• quantum physics
• system

• Ben-Gurion University of the Negev, Beer Sheva, Israel

  M. S. Livšic, N. Kravitsky, A. S. Markus

• Weizmann Institute of Science, Rehovot, Israel

  V. Vinnikov

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