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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Ind eed, it does not reveal the reason s for what ever distributive lattice t o be called an algebra since the term "algebra" refers cust omarily to conventional obj ects (d. ). The arising ambiguity is easily eliminated becau se a Boolean algebra is in fact an algebra over t he two-element field. At t he sa me time, it is perfectly natural to view Boolean algebras in different context s at differen t angles. It is worth emphasizing t hat t he particular Boolean algebr as we deal with in functional ana lysis appear mostly as distribut ive complemente d lattices.

1 (4)) . 4). 8) . 2) . If an LNS is simulta neously a vector lat tice t hen t here is another disjoint ness relat ion connected wit h it s lat tice structure. 9) . 10) . 1). 5) . e. 1). 2). 9). 3) are most frequent in ana lysis. 7) . 10). T hus, an LNS admits some functi on al represent ati on . A. G. 6) . 11) . 13) . The notion of measurable Banach bundle (MBB) is a measurable analog of a continuity structure. 1) . 3) . 5). Measurable Banach bundles with lifting are called liftable. There is a constructive connection between liftable measurable Banach bundles and complete continuous Banach bundles.

2) A space E is a universally com plete K -space (K-space of bounded elements) if and only if th e image of E und er the a bove isomorphism is th e whol e of Coo(Q) (th e subspace C (Q) of all continuous function s on Q) . (3) Every universally com plete K -spece E ca n be endowed with a multiplicat ion m akin g E into a faithful comm uta ti ve f -algebra. Moreover, if we require the order-uni ty fixed in E to be the multiplicati on unity, then this multiplication in E is unique. (4) Let D be a subset of a K-space E with an upper bound e E E .

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