Linear Operators: General theory |
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Page 304
... positive measure space and let = E - 18k , where fi , gk are positive elements in L , ( S , E , μ ) . Then there are positive elements hi in L2 ( S , Σ , μ ) , j = 1 , 2 , k = 1 , .. , n , such that n fj = Σ hjk , gk = h1k + hak j - 1 ...
... positive measure space and let = E - 18k , where fi , gk are positive elements in L , ( S , E , μ ) . Then there are positive elements hi in L2 ( S , Σ , μ ) , j = 1 , 2 , k = 1 , .. , n , such that n fj = Σ hjk , gk = h1k + hak j - 1 ...
Page 305
... positive elements . Let f , e L ( S , E , μ ) , f ; ≥ 0 , j = 1 , 2 , -1821 be decompositions of f1 and f2 into g1i + Σg2 ; is a decomposition of f1 + f2 , and let f1 = -161i f2 = fi 12 positive functions . Then and so To ( fi + f2 ) ...
... positive elements . Let f , e L ( S , E , μ ) , f ; ≥ 0 , j = 1 , 2 , -1821 be decompositions of f1 and f2 into g1i + Σg2 ; is a decomposition of f1 + f2 , and let f1 = -161i f2 = fi 12 positive functions . Then and so To ( fi + f2 ) ...
Page 714
... positive measure space . Suppose that T is a positive linear operator in L1 ( S , E , μ ) such that Th / n converges to zero in the weak operator topology and let | T ” —K | < 1 for some positive integer n and some compact operator K ...
... positive measure space . Suppose that T is a positive linear operator in L1 ( S , E , μ ) such that Th / n converges to zero in the weak operator topology and let | T ” —K | < 1 for some positive integer n and some compact operator K ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ