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.
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