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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**Example text**

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. 9. 10) . 13) . 1.

It is easily seen that each of t he sequences (X3n), (x3n-d, and (X3n-2) is pairwis e disjoint . By assumption there exist Yk := sUPnEN X3n-k (k :=0,1,2) . PutY :=YO VYI VY2 . Obviously, Yk(Q \elG) = {O} (k :=0,1,2),and therefore y(Q \ elG) = {O} . At the same time y(G) = {I} . Since Y is cont inuous, we have y( el G) = {I}. Thus, Y = Xc! G and el G is elopen set . Now, suppose that C(Q) is disjointly complete. Then Q is quasiextremal as we have just proved. Moreover, the Boolean algebras Clop(Q) a nd (1;(1) are isomorphic; therefore, Q is the Stone space of the Boolean algebra of elopen sets Clop(Q) .

An order interva l in E is a set of t he form [a, bJ := {x E E : a :S x :S b} , where a, b EE . The following constant ly used proper ty of vector lat t ices is frequ ently ca lled t he R iesz Decomp osit ion Property. (13 ) [0, x + y] = [0, x l + [0, yJ (x , Y E E+); We indicate only two corollaries of (13) . + X n /\ Y ( X k , Y E E +); /\ ~=l 2:;:1 X k ,l :S 2:j E J X l ,j ( l) /\ ... /\ Xn, j (n ) (Xl E E + and J is t he set of a ll funct ions j : {I , . . , n } -> {I, . . , m} . 3. T wo elements x a nd yare ca lled disjoint if Ixi/\ ly l = O.