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Lectures on Operator Theory and Its Applications
Edited by: Peter Lancaster, University of Calgary, AB, Canada
A co-publication of the AMS and Fields Institute.
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Fields Institute Monographs
1996; 339 pp; hardcover
Volume: 3
ISBN-10: 0-8218-0457-X
ISBN-13: 978-0-8218-0457-5
List Price: US$120
Member Price: US$96
Order Code: FIM/3
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Much of the importance of mathematics lies in its ability to provide theories which are useful in widely different fields of endeavor. A good example is the large and amorphous body of knowledge known as "the theory of linear operators" or "operator theory", which came to life about a century ago as a theory to encompass properties common to matrix, differential, and integral operators. Thus, it is a primary purpose of operator theory to provide a coherent body of knowledge which can explain phenomena common to the enormous variety of problems in which such linear operators play a part. The theory is a vital part of "functional analysis", whose methods and techniques are one of the major advances of twentieth century mathematics and now play a pervasive role in the modeling of phenomena in probability, imaging, signal processing, systems theory, etc., as well as in the more traditional areas of theoretical physics and mechanics.

This book is based on lectures presented at a meeting on operator theory and its applications held at the Fields Institute in the fall of 1994. The purpose of the meeting was to provide introductory lectures on some of the methods being used and problems being tackled in current research involving operator theory.

Titles in this series are co-published with The Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

Readership

Research mathematicians.

Table of Contents

  • Lecture Series 1. A. Böttcher, Infinite matrices and projection methods
  • Matrix representation of operators
  • Three problems for infinite matrices
  • The finite section method
  • Selfadjoint and compact opeators
  • Toeplitz matrices with continuous symbols
  • Toeplitz operators: algebraization of stability
  • Toeplitz operators: localization
  • Block case and higher dimensions
  • Banach space phenomena
  • Norms of inverses and pseudospectra
  • Toeplitz determinants
  • More general projection methods
  • Bibliography
  • Lecture Series 2. A. Dijksma and H. Langer, Operator theory and ordinary differential operators
  • Introduction
  • Definitizable operators in Kreĭn spaces
  • Boundary eigenvalue problems for Sturm-Liouville operators and related holomorphic functions
  • Operator representations of holomorphic functions
  • Sturm-Liouville operators with indefinite weight
  • Interface conditions and singular potentials
  • Operator pencils
  • Bibliography
  • Symbols used in the lecture
  • Lecture Series 3. M. A. Dritschel and J. Rovnyak, Operators on indefinite inner product spaces
  • Introduction: Preliminaries and notation
  • Kreĭn spaces and operators
  • Julia operators and contractions
  • Extension and completion problems
  • The Schur algorithm
  • Reproducing kernel Pontryagin spaces and colligations
  • Invariant subspaces
  • Bibliography
  • Lecture Series 4. M. A. Kaashoek, State space theory of rational matrix functions and applications
  • Introduction
  • Canonical factorization and the state space method
  • \(J\)-unitary rational matrix functions
  • Analysis of zeros
  • Inverse problems involving null pairs
  • Analysis of zeros and poles
  • Inverse problems involving null-pole triples
  • Bibliography
  • Index
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