Linear Operators: General theory |
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Page 614
... semi - group of operators in X ; i.e. , a family of operators satisfying the conditions of the following definition . 1 DEFINITION . A family { T ( t ) } , 0 ≤t < ∞ , of bounded linear operators in X will be called a strongly continuous ...
... semi - group of operators in X ; i.e. , a family of operators satisfying the conditions of the following definition . 1 DEFINITION . A family { T ( t ) } , 0 ≤t < ∞ , of bounded linear operators in X will be called a strongly continuous ...
Page 655
... semi- group on [ 0 , ∞ ) . Equivalently , T ( t ) ( x , s ) = Σ∞∞∞ne¬n2 + ins if x ( s ) eins . Show that the infinitesimal generator of the semi- group T ( t ) is the operator A whose domain D ( 4 ) consists of all periodic ...
... semi- group on [ 0 , ∞ ) . Equivalently , T ( t ) ( x , s ) = Σ∞∞∞ne¬n2 + ins if x ( s ) eins . Show that the infinitesimal generator of the semi- group T ( t ) is the operator A whose domain D ( 4 ) consists of all periodic ...
Page 697
... semi - group of operators in L1 ( S , Σ , μ ) with \ T ( t1 , ... , tx ) ≤ 1 , T ( t , ... , tx ) ∞ ≤1 . Let 1 ≤ p < ∞ , fe Lp , and f * ( s ) 1 α A ( a ) = ak 0 1 . 11 = sup A ( x ) ( f , s ) where 8 < 8 < ∞ [ * T ( ty , . . . , tx ) ...
... semi - group of operators in L1 ( S , Σ , μ ) with \ T ( t1 , ... , tx ) ≤ 1 , T ( t , ... , tx ) ∞ ≤1 . Let 1 ≤ p < ∞ , fe Lp , and f * ( s ) 1 α A ( a ) = ak 0 1 . 11 = sup A ( x ) ( f , s ) where 8 < 8 < ∞ [ * T ( ty , . . . , tx ) ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space 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 Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ