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 ) = Σ∞∞ ‰ ¤‚e¬n3t + 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∞∞ ‰ ¤‚e¬n3t + 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 L ( S , E , μ ) with T ( t1 , ... , tx ) 1 ≤ 1 , T ( t , ... , tx ) ∞ ≤ 1 . Let 1 ≤ p < ∞ , fe Lp , and f * ( s ) = sup 4 ( a ) ( f , s ) where 1 α A ( a ) = ak 0 0 < α < ∞ ... S T ( t tk ) dt . dtx ...
... semi - group of operators in L ( S , E , μ ) with T ( t1 , ... , tx ) 1 ≤ 1 , T ( t , ... , tx ) ∞ ≤ 1 . Let 1 ≤ p < ∞ , fe Lp , and f * ( s ) = sup 4 ( a ) ( f , s ) where 1 α A ( a ) = ak 0 0 < α < ∞ ... S T ( t tk ) dt . dtx ...
Contents
B Topological Preliminaries | 10 |
Metric Spaces | 23 |
Product Spaces | 31 |
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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 Doklady Akad Duke Math E₁ elements ergodic exists extension f₁ function defined function f Hausdorff space Hence Hilbert space homomorphism inequality 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 o-field 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 topological space u-integrable u-measurable u-null uniformly unit sphere valued function vector space weakly compact zero ΕΕΣ