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Principal Currents for a Pair of Unitary Operators
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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

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.

• 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