Theory of Commuting Nonselfadjoint Operators

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

Part of the Mathematics and Its Applications book series (MAIA, volume 332)

Table of contents

  1. Front Matter
    Pages i-xvii
  2. operator Vessels in Hilbert Space

    1. Front Matter
      Pages 1-1
    2. M. S. Livšic, N. Kravitsky, A. S. Markus, V. Vinnikov
      Pages 3-17
    3. M. S. Livšic, N. Kravitsky, A. S. Markus, V. Vinnikov
      Pages 18-28
    4. M. S. Livšic, N. Kravitsky, A. S. Markus, V. Vinnikov
      Pages 29-41
    5. 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
    2. M. S. Livšic, N. Kravitsky, A. S. Markus, V. Vinnikov
      Pages 73-80
    3. M. S. Livšic, N. Kravitsky, A. S. Markus, V. Vinnikov
      Pages 81-91
    4. 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
    2. M. S. Livšic, N. Kravitsky, A. S. Markus, V. Vinnikov
      Pages 103-130
    3. 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
    2. M. S. Livšic, N. Kravitsky, A. S. Markus, V. Vinnikov
      Pages 187-233
    3. M. S. Livšic, N. Kravitsky, A. S. Markus, V. Vinnikov
      Pages 234-255
    4. M. S. Livšic, N. Kravitsky, A. S. Markus, V. Vinnikov
      Pages 256-301
  6. Back Matter
    Pages 303-318

About this book

Introduction

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.

Keywords

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

Authors and affiliations

  • M. S. Livšic
    • 1
  • N. Kravitsky
    • 1
  • A. S. Markus
    • 1
  • V. Vinnikov
    • 2
  1. 1.Ben-Gurion University of the NegevBeer ShevaIsrael
  2. 2.Weizmann Institute of ScienceRehovotIsrael

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-015-8561-3
  • Copyright Information Springer Science+Business Media B.V. 1995
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-90-481-4585-0
  • Online ISBN 978-94-015-8561-3
  • About this book