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Linear functional equations. An operator approach. (Linejnye funktsional’nye uravneniya. Operatornyj podkhod.) (Russian) Zbl 0708.47010

Minsk: Izdatel’stvo “Universitetskoe”. 232 p. R. 2.00 (1988).
The main purpose of the book is to investigate functional operators \[ (1)\quad Au(x)\equiv \sum^{n}_{1}a_ k(x)u(\alpha_ k(x)),\quad \alpha_ k(x): X\to X,\quad x\in X,\quad k=1,2,...,n; \] namely the spectral properties, invertibility, Fredholmity and the index of such operators are described. Some applications of the obtained results are demonstrated in Chapter 5. Operators (1) are very general and their investigation requires some tools and results from different fields of mathematics. Some of this material is exposed in Chapter 0. The book is based on the original investigations of the author during the last 20 years. Here is the content, which gives the right impression about the book.
Introduction.
Chapter 0. Preliminary information.
Chapter 1. Functional operators.
§ 1. Operators of a weighted shift. Conditions for the boundedness.
§ 2. Operator approach. Model examples.
§ 3. Discussing the model examples. Discontinuity of the spectral radius.
§ 4. Operators generated by the finite group of transformations.
§ 5. Spectral radius of a weighted shift operator.
Chapter 2. Banach algebras, generated by functional operators.
§ 6. Extension of functional algebras with the help of operators, generating automorphisms.
§ 7. Theorems on the isomorphism of \(C^*\)-algebras, generated by dynamical systems.
§ 8. Consequences of the theorem on isomorhism.
Chapter 3. Invertibility conditions of functional operators. \(L_ 2\)- theory.
§ 9. Characterization of the invertibility with the help of hyperbolic linear extensions.
§ 10. Discretization of the trajectory approach to the investigation of the invertibility.
§ 11. Operators with convex rationally independent system of shifts.
§ 12. Algebras of type \(C^*(A,G,T_ g)\) for a free acting group.
Chapter 4. Functional operators for special spaces of functions.
§ 13. Operators of a weighted shift in spaces of smooth functions.
§ 14. Functional equations in the space of periodic distributions.
Chapter 5.
§ 15. Ordinary functional-differential equations of the neutral type.
§ 16. Symbol of a partial functional-differential operator.
§ 17. Non-local boundary value problems for elliptic equations.
§ 18. Equations with periodic and almost periodic coefficients.
§ 19. Boundary value problems with data over the whole boundary for the string oscillation equation.
§ 20. Index formula.
Commentaries and the bibliographical information.
References.
Reviewer: R.Dudučava

MSC:

47A50 Equations and inequalities involving linear operators, with vector unknowns
47B38 Linear operators on function spaces (general)
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
47A10 Spectrum, resolvent
47A53 (Semi-) Fredholm operators; index theories
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