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.
