# Dominated Operators by A.G. Kusraev

By A.G. Kusraev

The proposal of a ruled or rnajorized operator rests on an easy concept that is going way back to the Cauchy approach to majorants. Loosely talking, the assumption might be expressed as follows. If an operator (equation) lower than learn is ruled by means of one other operator (equation), known as a dominant or majorant, then the houses of the latter have a considerable effect at the homes of the previous . therefore, operators or equations that experience "nice" dominants needs to own "nice" houses. In different phrases, an operator with a someway certified dominant needs to be certified itself. Mathematical instruments, placing the assumption of domination right into a common and whole shape, have been urged via L. V. Kantorovich in 1935-36. He brought the funda­ psychological thought of a vector area normed through components of a vector lattice and that of a linear operator among such areas that is ruled through a good linear or monotone sublinear operator. He additionally utilized those notions to fixing sensible equations. within the succeedingyears many authors studied quite a few specific circumstances of lattice­ normed areas and varied sessions of ruled operators. although, examine was once played inside and within the spirit of the idea of vector and normed lattices. So, it's not an exaggeration to claim that ruled operators, as autonomous items of research, have been past the succeed in of experts for part a century. subsequently, an important structural homes and a few fascinating purposes of ruled operators became on hand considering the fact that recently.

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