Linear Functional Equations: Operator Approach

TL;DR: In this paper, the spectral radius of a Weighted shift operator is analyzed in the context of boundary value problems for elliptic and almost periodic functions, and the symbol of a functional-partial differential operator is defined.

Abstract: 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.