eBook ISBN:  9781470400996 
Product Code:  MEMO/109/522.E 
List Price:  $40.00 
MAA Member Price:  $36.00 
AMS Member Price:  $24.00 
eBook ISBN:  9781470400996 
Product Code:  MEMO/109/522.E 
List Price:  $40.00 
MAA Member Price:  $36.00 
AMS Member Price:  $24.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 109; 1994; 103 ppMSC: Primary 47; 46
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.
ReadershipOperator theorists, functional analysts and possibly graduate students.

Table of Contents

Chapters

0. Introduction

1. The geometry associated with eigenvalues

2. The dilation space solution of the symbol Riemann Hilbert problem

3. The principal current for the operatortuple $\{P_1, P_2, W_1, W_2\}$

4. Estimates

5. The criterion for eigenvalues

6. The $N(\omega )$ operator

7. The characteristic operator function of $T_1$

8. Localization and the “cutdown” property

9. The joint essential spectrum

10. Singular integral representations

11. Toeplitz operators with unimodular symbols

12. $C_{11}$contraction operators with (1,1) deficiency indices


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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 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.
Operator theorists, functional analysts and possibly graduate students.

Chapters

0. Introduction

1. The geometry associated with eigenvalues

2. The dilation space solution of the symbol Riemann Hilbert problem

3. The principal current for the operatortuple $\{P_1, P_2, W_1, W_2\}$

4. Estimates

5. The criterion for eigenvalues

6. The $N(\omega )$ operator

7. The characteristic operator function of $T_1$

8. Localization and the “cutdown” property

9. The joint essential spectrum

10. Singular integral representations

11. Toeplitz operators with unimodular symbols

12. $C_{11}$contraction operators with (1,1) deficiency indices