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Full Description
In this book we shall study linear functional equations of the form m bu(x) == Lak(X)U(Qk(X)) = f(x), (1) k=l where U is an unknown function from a given space F(X) of functions on a set X, Qk: X -+ X are given mappings, ak and f are given functions. Our approach is based on the investigation of the operators given by the left-hand side of equa tion (1). In what follows such operators will be called functional operators. We will pay special attention to the spectral properties of functional operators, first of all, to invertibility and the Noether property. Since the set X, the space F(X), the mappings Qk and the coefficients ak are arbitrary, the class of operators of the form (1) is very rich and some of its individ ual representatives are related with problems arising in various areas of mathemat ics and its applications. In addition to the classical theory of functional equations, among such areas one can indicate the theory of functional-differential equations with deviating argument, the theory of nonlocal problems for partial differential equations, the theory of boundary value problems for the equation of a vibrating string and equations of mixed type, a number of problems of the general theory of operator algebras and the theory of dynamical systems, the spectral theory of au tomorphisms of Banach algebras, and other problems.
Contents
0. Introductory Material.- 1. Functional Operators.- 1. Weighted Shift Operators. Boundedness Conditions.- 2. The Operator Approach. Model Examples.- 3. Discussion of the Model Examples Discontinuity of the Spectral Radius.- 4. Operators Generated by a Finite Transformation Group.- 5. Spectral Radius of a Weighted Shift Operator.- 2. Banach Algebras Generated by Functional Operators.- 6. Extension of Operator Algebras by Operators that Generate Automorphisms.- 7. The Isomorphism Theorem for C* -Algebras Generated by Dynamical Systems.- 8. Corollaries of the Isomorphism Theorem.- 3. Invertibility Conditions for Functional Operators. L2-Theory.- 9. Characterization of Invertible Operators by Means of Hyperbolic Linear Extensions.- 10. Discretization as an Orbital Approach to Study Invertibility.- 11. Operators with a Convex Rationally Independent System of Shifts.- 12. Algebras of Type C*(A, G, Tg) in the Case of an Arbitrary Group Action.- 4. Functional Operators in Some Special Function Spaces.- 13. Weighted Shift Operators in Spaces of Smooth Functions.- 14. Functional Equations in a Space of Periodic Distributions.- 5. Applications to Some Classes of Equations and Boundary Value Problems.- 15. Functional-Differential Equations of Neutral Type.- 16. The Symbol of a Functional-Partial Differential Operator.- 17. Nonlocal Boundary Value Problems for Elliptic Equations.- 18. Equations with Periodic and Almost Periodic Coefficients.- 19. Boundary Value Problems with Data on the Entire Boundary for the Equation of a Vibrating String.- 20. Index Formulas.- Comments and Bibliographic Information.- References.
