Linear Operators: General theory |
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Page 630
... semi - group . Theorem 13 gives necessary and sufficient conditions , but these conditions are often difficult to verify in concrete analytical cases of interest . Consequently , we shall devote the rest of this section to an ...
... semi - group . Theorem 13 gives necessary and sufficient conditions , but these conditions are often difficult to verify in concrete analytical cases of interest . Consequently , we shall devote the rest of this section to an ...
Page 689
... semi - group of operators in L1 ( S , Σ , μ ) which is strongly integrable on every finite interval . It is assumed that for some constant K | A ( α ) | 1 ≤K , | T ( t ) ∞ ≤K , 0 < a , t , and that T ( n ) / n converges to zero ...
... semi - group of operators in L1 ( S , Σ , μ ) which is strongly integrable on every finite interval . It is assumed that for some constant K | A ( α ) | 1 ≤K , | T ( t ) ∞ ≤K , 0 < a , t , and that T ( n ) / n converges to zero ...
Page 697
... semi - group of operators in L ( S , E , μ ) with T ( t1 , ... , tk ) 1 ≤ 1 , T ( t1 , ... , tr ) ∞ ≤1 . Let 1 ≤ p < ∞o , fe Lp , and f * ( s ) sup 4 ( a ) ( f , s ) where - 8 < 8 < ∞ 1 να A ( x ) = ― - ୮ ST ( 4 , ... tx ) dt ...
... semi - group of operators in L ( S , E , μ ) with T ( t1 , ... , tk ) 1 ≤ 1 , T ( t1 , ... , tr ) ∞ ≤1 . Let 1 ≤ p < ∞o , fe Lp , and f * ( s ) sup 4 ( a ) ( f , s ) where - 8 < 8 < ∞ 1 να A ( x ) = ― - ୮ ST ( 4 , ... tx ) dt ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ