Linear functional equations. Operator approach. Transl. from the Russian by V. Muzafarov a. A. Iacob
type: Item
instance of: scholarly article
title: Linear functional equations. Operator approach. Transl. from the Russian by V. Muzafarov a. A. Iacob
zbMATH Open document ID: 0841.47001
published in: Operator Theory: Advances and Applications
publication date: 1996-02-08T00:00:00Z
review text: 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 wel…
reviewed by: N. Karapetiants
Mathematics Subject Classification ID: 47-02
Mathematics Subject Classification ID: 47A50
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47G30
zbMATH DE Number: 843247
zbMATH Keywords: functional equation
zbMATH Keywords: singular or convolution type operators
zbMATH Keywords: Noether property
zbMATH Keywords: invertibility
zbMATH Keywords: operator algebras
zbMATH Keywords: dynamical systems
zbMATH Keywords: mixed type problems
zbMATH Keywords: symbol
zbMATH Keywords: index
zbMATH Keywords: isomorphism between \(C^*\)-algebras
zbMATH Keywords: functional or pseudodifferential equations with shift
zbMATH Keywords: spectral properties
author: A. B. Antonevich
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 843247