Linear Operators: General theory |
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Page 685
... semi - group upon the parameter t . We recall that the semi - group { T ( t ) , 0 ≤ t } is said to be strongly con- tinuous if its dependence upon t is continuous in the strong operator topology , i.e. , if lim T ( t ) x - T ( u ) x ...
... semi - group upon the parameter t . We recall that the semi - group { T ( t ) , 0 ≤ t } is said to be strongly con- tinuous if its dependence upon t is continuous in the strong operator topology , i.e. , if lim T ( t ) x - T ( u ) x ...
Page 689
... semi - group of operators in L1 ( S , E , μ ) which is strongly integrable on every finite interval . It is assumed that for some constant K 0 < a , t , and that T ( n ) / n converges to zero strongly in L1 ( S , Σ , μ ) . Then the ...
... semi - group of operators in L1 ( S , E , μ ) which is strongly integrable on every finite interval . It is assumed that for some constant K 0 < a , t , and that T ( n ) / n converges to zero strongly in L1 ( S , Σ , μ ) . Then the ...
Page 697
... semi - group of operators in L1 ( S , Σ , μ ) with T ( t , . . . , tx ) 1 ≤ 1 , T ( t1 , . . . , tx ) | ∞ ≤ 1 . Let 1 ≤ p < ∞ , fe Lp , and f * ( s ) A ( x ) = 1 - ak = ( ... ( * Th 0 sup 4 ( a ) ( f , s ) where 8 < 8 < ∞ [ * T ...
... semi - group of operators in L1 ( S , Σ , μ ) with T ( t , . . . , tx ) 1 ≤ 1 , T ( t1 , . . . , tx ) | ∞ ≤ 1 . Let 1 ≤ p < ∞ , fe Lp , and f * ( s ) A ( x ) = 1 - ak = ( ... ( * Th 0 sup 4 ( a ) ( f , s ) where 8 < 8 < ∞ [ * T ...
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 converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f g₁ 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-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ