# Dominated Operators by A.G. Kusraev

By A.G. Kusraev

The concept of a ruled or rnajorized operator rests on an easy concept that is going way back to the Cauchy approach to majorants. Loosely conversing, the belief could be expressed as follows. If an operator (equation) below examine is ruled via one other operator (equation), referred to as a dominant or majorant, then the houses of the latter have a considerable impact at the homes of the previous . therefore, operators or equations that experience "nice" dominants needs to own "nice" homes. In different phrases, an operator with a by some means certified dominant has to be certified itself. Mathematical instruments, placing the assumption of domination right into a normal and whole shape, have been instructed through L. V. Kantorovich in 1935-36. He brought the funda­ psychological idea of a vector area normed by way of parts of a vector lattice and that of a linear operator among such areas that's ruled by way of a good linear or monotone sublinear operator. He additionally utilized those notions to fixing useful equations. within the succeedingyears many authors studied a variety of specific instances of lattice­ normed areas and diversified sessions of ruled operators. besides the fact that, examine was once played inside of and within the spirit of the idea of vector and normed lattices. So, it isn't an exaggeration to assert that ruled operators, as self sufficient gadgets of research, have been past the succeed in of experts for part a century. as a result, an important structural homes and a few fascinating purposes of ruled operators became to be had for the reason that recently.

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