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Principal Currents for a Pair of Unitary Operators
Joel D. Pincus and Shaojie Zhou

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$40
Individual Members: US$24
Institutional Members: US$32
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.


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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