Linear functional equations. Operator approach. Transl. from the Russian by V. Muzafarov a. A. Iacob
This book is the English translation of the Russian original published in 1988. The main subject of this book in the functional equation with shift of the form \[ (bu)(x)= \sum^m_{k= 1} a_k(x) u(\alpha_k(x))= f(x). \] Here \(\alpha_k: X\to X\) are given mappings on a given set \(X\) and the coefficients \(a_k\), the right part \(f(x)\) and the unknown function \(u(x)\) belong to the same space \(F(X)\). The operator \(b\) in this book, as a rule, is a part of the general operator (or general algebra of operators) containing, for example, also singular or convolution type operators, etc. The main attention is paid to the connection with problems of Noether property, invertibility and spectral properties. Since \(X\), \(F(X)\), \(a_k\), \(\alpha_k\) are arbitrary, many interesting problems from various areas of mathematics and physics may be described in terms containing the operator \(b\). There are problems from the theory of operator algebras, from dynamical systems, mixed type problems, etc. Contents: The book consists of six chapters: Chapter 0. Introduction material. Chapter 1. Functional operators. Chapter 2. Banach algebras generated by functional operators. Chapter 3. Invertibility conditions for functional operators. \(L_2\)-theory. Chapter 4. Functional operators in some special function spaces. Chapter 5. Applications to some classes of equations and boundary value problems. Comments and bibliographic information. The author gives many interesting examples and considers many model operators connected with the operator \(b\) in the space \(L_2(X)\) as well as in the spaces of smooth functions. He describes the spectral properties of these model operators. The chapters 1-3 contain the main results of this book particularly about Noether properties, symbol, index, and isomorphism between \(C^*\)-algebras and some dynamical systems. Chapter 5 contains some applications to functional or pseudodifferential equations with shift. This interesting and extremely well written book may be useful to a broad circle of specialists in operator theory, in the theory of dynamical systems, in the theory of differential and integral equations and also for students.
functional or pseudodifferential equations with shift
scientific article; zbMATH DE number 843247