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Principal Currents for a Pair of Unitary Operators
Joel D. Pincus and Shaojie Zhou
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Memoirs of the American Mathematical Society
1994; 103 pp; softcover
Volume: 109
ISBN-10: 0-8218-2609-3
ISBN-13: 978-0-8218-2609-6
List Price: US$38
Individual Members: US$22.80
Institutional Members: US$30.40
Order Code: MEMO/109/522
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Principal currents were invented to provide a noncommutative spectral theory in which there is still significant localization. These currents are often integral and are associated with a vector field and an integer-valued weight which plays the role of a multi-operator index. The study of principal currents involves scattering theory, new geometry associated with operator algebras, defect spaces associated with Wiener-Hopf and other integral operators, and the dilation theory of contraction operators. This monograph explores the metric geometry of such currents for a pair of unitary operators and certain associated contraction operators. Applications to Toeplitz, singular integral, and differential operators are included.

Readership

Operator theorists, functional analysts and possibly graduate students.

Table of Contents

  • Introduction
  • The geometry associated with eigenvalues
  • The dilation space solution of the symbol Riemann Hilbert problem
  • The principal current for the operator-tuple \(\{P_1, P_2, W_1, W_2\}\)
  • Estimates
  • The criterion for eigenvalues
  • The \(N(\omega )\) operator
  • The characteristic operator function of \(T_1\)
  • Localization and the "cut-down" property
  • The joint essential spectrum
  • Singular integral representations
  • Toeplitz operators with unimodular symbols
  • \(C_{11}\)-Contraction operators with \((1,1)\) deficiency indices
  • Appendix A
  • Appendix B
  • Appendix C
  • References
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