Linear Operators: General theory |
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Page 613
... semi - group of operators . In this section will be found the salient features of the analytical theory of these semi - groups as developed by E. Hille , R. S. Phillips , and K. Yosida . By analogy with the scalar case we could expect that ...
... semi - group of operators . In this section will be found the salient features of the analytical theory of these semi - groups as developed by E. Hille , R. S. Phillips , and K. Yosida . By analogy with the scalar case we could expect that ...
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 ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional 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 Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ