# Principal Currents for a Pair of Unitary Operators

Share this page
*Joel D. Pincus; Shaojie Zhou*

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.

#### Table of Contents

# Table of Contents

## Principal Currents for a Pair of Unitary Operators

- Contents v6 free
- §0. Introduction 18 free
- §1. The geometry associated with eigenvalues 916 free
- §2. The dilation space solution of the symbol Riemann Hilbert problem 1926
- §3. The principal current for the operator-tuple {P[sub(1)], P[sub(2)], W[sub(1)],W[sub(2)]} 2633
- §4. Estimates 3340
- §5. The criterion for eigenvalues 4047
- §6. The N(w) operator 4451
- §7. The characteristic operator function of T[sub(1)] 5259
- §8. Localization and the "cut-down" property 6067
- §9. The joint essential spectrum 6673
- §10. Singular integral representations 7481
- §11. Toeplitz operators with unimodular symbols 8087
- §12. C[sub(11)]-Contraction operators with (1,1) deficiency indices 8491
- §13. Appendix A 9299
- §14. Appendix B 97104
- §15. Appendix C 98105
- References 100107