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 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 will be called a strongly continuous ...
Page 655
... semi- group on [ 0 , ∞ ) . Equivalently , T ( t ) ( x , s ) = Σane - 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 functions ...
... semi- group on [ 0 , ∞ ) . Equivalently , T ( t ) ( x , s ) = Σane - 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 functions ...
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 ...
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