Linear Operators: General theory |
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Page 630
... semi - group . Theorem 13 gives necessary and sufficient conditions , but these conditions are often difficult to verify in concrete analytical cases of interest . Consequently , we shall devote the rest of this section to an ...
... semi - group . Theorem 13 gives necessary and sufficient conditions , but these conditions are often difficult to verify in concrete analytical cases of interest . Consequently , we shall devote the rest of this section to an ...
Page 689
... semi - group of operators in L1 ( S , Z , μ ) which is strongly integrable on every finite interval . It is assumed that for some constant K 4 ( a ) , ≤K , T ( t ) ∞ ≤K , 0 < a , t , and that T ( n ) / n converges to zero strongly in ...
... semi - group of operators in L1 ( S , Z , μ ) which is strongly integrable on every finite interval . It is assumed that for some constant K 4 ( a ) , ≤K , T ( t ) ∞ ≤K , 0 < a , t , and that T ( n ) / n converges to zero strongly in ...
Page 697
... semi - group of operators in L1 ( S , E , μ ) with T ( t ,, ... , tz ) 1 ≤ 1 , T ( t , ... , tx ) | ∞ ≤ 1 . Let 1 ≤ p < ∞ , fe L ,, and f * ( s ) = sup | 4 ( a ) ( f , s ) where 1 τα A ( α ) = ak κα 8 < x < ∞ tz ) dt1 [ ... [ * T ( ...
... semi - group of operators in L1 ( S , E , μ ) with T ( t ,, ... , tz ) 1 ≤ 1 , T ( t , ... , tx ) | ∞ ≤ 1 . Let 1 ≤ p < ∞ , fe L ,, and f * ( s ) = sup | 4 ( a ) ( f , s ) where 1 τα A ( α ) = ak κα 8 < x < ∞ tz ) dt1 [ ... [ * T ( ...
Contents
Preliminary Concepts | 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 ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element ergodic exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f 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 S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ