# Dominated Operators

• Anatoly G. Kusraev
Book

Part of the Mathematics and Its Applications book series (MAIA, volume 519)

1. Front Matter
Pages i-xiii
2. Anatoly G. Kusraev
Pages 1-43
3. Anatoly G. Kusraev
Pages 44-88
4. Anatoly G. Kusraev
Pages 89-140
5. Anatoly G. Kusraev
Pages 141-186
6. Anatoly G. Kusraev
Pages 187-235
7. Anatoly G. Kusraev
Pages 236-290
8. Anatoly G. Kusraev
Pages 291-337
9. Anatoly G. Kusraev
Pages 338-393
10. Back Matter
Pages 394-446

### Introduction

The notion of a dominated or rnajorized operator rests on a simple idea that goes as far back as the Cauchy method of majorants. Loosely speaking, the idea can be expressed as follows. If an operator (equation) under study is dominated by another operator (equation), called a dominant or majorant, then the properties of the latter have a substantial influence on the properties of the former . Thus, operators or equations that have "nice" dominants must possess "nice" properties. In other words, an operator with a somehow qualified dominant must be qualified itself. Mathematical tools, putting the idea of domination into a natural and complete form, were suggested by L. V. Kantorovich in 1935-36. He introduced the funda­ mental notion of a vector space normed by elements of a vector lattice and that of a linear operator between such spaces which is dominated by a positive linear or monotone sublinear operator. He also applied these notions to solving functional equations. In the succeedingyears many authors studied various particular cases of lattice­ normed spaces and different classes of dominated operators. However, research was performed within and in the spirit of the theory of vector and normed lattices. So, it is not an exaggeration to say that dominated operators, as independent objects of investigation, were beyond the reach of specialists for half a century. As a consequence, the most important structural properties and some interesting applications of dominated operators have become available since recently.

### Keywords

functional functional analysis integration theory mathematical logic vector lattice

#### Authors and affiliations

• Anatoly G. Kusraev
• 1
• 2
1. 1.Institute for Applied Mathematics and InformaticsNorth Ossetian State UniversityVladikavkazRussia
2. 2.Sobolev Institute of Mathematics, Siberian DivisionRussian Academy of SciencesNovosibirskRussia

### Bibliographic information

• DOI https://doi.org/10.1007/978-94-015-9349-6