Linear Operators: General theory |
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Page 304
... positive and write T≥ 0 when ƒ e L , and ƒ ≥ 0 imply that Tf ≥ 0. Similarly T1 ≥ T2 or T2 ≤ T1 means that T1 ... positive measure space and let 1 ≤ p ≤∞ . Suppose f1 + f2 = Σn - 18 , where ƒj , g1 are positive elements in L , ( S ...
... positive and write T≥ 0 when ƒ e L , and ƒ ≥ 0 imply that Tf ≥ 0. Similarly T1 ≥ T2 or T2 ≤ T1 means that T1 ... positive measure space and let 1 ≤ p ≤∞ . Suppose f1 + f2 = Σn - 18 , where ƒj , g1 are positive elements in L , ( S ...
Page 305
... positive elements . Let f , e L ( S , E , μ ) , f ; ≥ 0 , j = 1 , 2 , n and let f1 = -161i f2 = -182i be decompositions of f1 and f2 into fi positive functions . Then 81 + 2 is a decomposition of f1 + f2 , and so To ( fi + f2 ) ≤ To ...
... positive elements . Let f , e L ( S , E , μ ) , f ; ≥ 0 , j = 1 , 2 , n and let f1 = -161i f2 = -182i be decompositions of f1 and f2 into fi positive functions . Then 81 + 2 is a decomposition of f1 + f2 , and so To ( fi + f2 ) ≤ To ...
Page 714
... positive measure space . Suppose that T is a positive linear operator in L1 ( S , Σ , μ ) such that Tr / 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 , Σ , μ ) such that Tr / 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 ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ