Functional Differential Equations: II. C*-Applications Part 1: Equations with Continuous Coefficients
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Summary
Together with the authors' Volume I. C*-Theory, the two parts comprising Functional Differential Equations: II. C*-Applications form a masterful work-the first thorough, up-to-date exposition of this field of modern analysis lying between differential equations and C*-algebras.
The two parts of Volume II contain the applications of the C*-structures and theory developed in Volume I. They show the technique of using the C*-results in the study of the solvability conditions of non-local functional differential equations and demonstrate the fundamental principles underlying the interrelations between C* and functional differential objects. The authors focus on non-local pseudodifferential, singular integral, and Toeplitz operators-with continuous and piecewise continuous coefficients-convolution type operators with oscillating coefficients and shifts, and operators associated with non-local boundary value problems containing transformation operators of an argument on the boundary. They build the symbolic calculus for all these classes of operators, use it to treat concrete examples of non-local operators, present the explicit computation of their Fredholmity conditions and the index formulae, and obtain a number of related results.
Part 1: Equations with Continuous Coefficients and Part 2: Equations with Discontinuous Coefficients and Boundary Value Problems can each stand alone and prove a valuable resource for researchers and students interested in operator algebraic methods in the theory of functional differential equations, and to pure C*-algebraists looking for important and promising new applications. Together these books form a powerful library for this intriguing field of modern analysis.
Table of Contents
PART 1:
Preface to Volume II
Introduction to Volume II
G-Pseudodifferential Operators with Continuous Coefficients
Symbolic Calculus
G-Pseudodifferential Operators on Rn. Pseudodifferential Difference Operators and Pseudodifferential Operators with Affine Shifts
The Measures on the Sphere Ergodic under the Induced Linear Actions
G-PDOs on Rn with "Linear" Shifts. Examples
The Symbolic Calculus of G-PDOs on a Manifold L2-Case)
The Symbolic Calculus of G-PDOs in Sobolev Spaces
The Symbolic Calculus of G-PDOs on Vector Bundles. The Main Theorem
G-PDOs with a Finite Group of Shifts
G-PDOs on the Torus. Examples
G-PDOs. Examples
G-Singular Integral Operators with Continuous Coefficients
The Index of a G-SIO
G-Toeplitz Operators with Continuous Coefficients
Boundary Value Problems with Strong Non-Localness
The Convolution Type Operators
Notes and Remarks
Appendices
The Index Calculation in C*(A,Tg)
Quasicross-Products and C*-Algebras Associated with Quasirepresentations of Groups
Notes and Remarks
Bibliography
Index
List of Notation
PART 2:
Preface
Preface to Volume II
G-Pseudodifferential Operators with Piecewise Continuous Coefficients
Symbolic Calculus of G-PDOs with Piecewise Continuous Coefficients
G-PDOs with Piecewise Continuous Coefficients having Singularities on a Singled Out Set
Algebras Generated by Singular Integral Operators with Piecewise Continuous Coefficients and Piecewise Smooth Mappings of a Curve
Algebras Generated by Teoplitz Operators with Piecewise Continuous Coefficients and Smooth Mappings of a Curve
Algebras Generate by Toeplitz Operators with Piecewise Continuous Coefficients and Piecewise Smooth Mappings of a Curve
The Index Calculation for G-SIOs and G-Toeplitz Operators with Piecewise Continuous Coefficients. Kveselava-Vekua Type Theorems
G-Boundary Value Problems
C*-Algebras Arising from the Wiener-Hopf Operators
Miscellaneous Results. PDOs with the Transmission Property
The Boutet de Monvel Operators of the Class B0
The Spectrum of the C*-Algebra of the Principal Symbols of Elements of B0
G-Boutet de Monvel Operators (G-BMOs)
G-Boundary Value Problem Operators (G-BVPOs) I
G-BVPOs II
Example
G-BVPOs III
Non-Local Agranovich-Dynin Type Theorem
Transmission Problems
G-BVPOs in Sobolev Spaces
The Localization Theorem. Reduction of G-BVPOs to G-BMOs
Notes and Remarks
Appendix C
C*-Algebras Associated with Automorphisms Acting on a Singled Out Ideal
Notes and Remarks
Bibliography
Index
List of Notation
