Memoirs of the American Mathematical Society 1994; 103 pp; softcover Volume: 109 ISBN10: 0821826093 ISBN13: 9780821826096 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 integervalued weight which plays the role of a multioperator index. The study of principal currents involves scattering theory, new geometry associated with operator algebras, defect spaces associated with WienerHopf 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 operatortuple \(\{P_1, P_2, W_1, W_2\}\)
 Estimates
 The criterion for eigenvalues
 The \(N(\omega )\) operator
 The characteristic operator function of \(T_1\)
 Localization and the "cutdown" 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
