Linear Operators: General theory |
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Page 603
... infinity . ( a ) If x is in D ( T " ) , then f ( T ) x is in D ( Tm + n ) , where m + n = ∞o if m = ∞ , and P ( T ) f ( T ) x f ( T ) P ( T ) x . ( b ) If 0 ≤ n ≤m , and g ( 2 ) g ( T ) = P ( T ) f ( T ) . = = P ( 2 ) f ( 2 ) ...
... infinity . ( a ) If x is in D ( T " ) , then f ( T ) x is in D ( Tm + n ) , where m + n = ∞o if m = ∞ , and P ( T ) f ( T ) x f ( T ) P ( T ) x . ( b ) If 0 ≤ n ≤m , and g ( 2 ) g ( T ) = P ( T ) f ( T ) . = = P ( 2 ) f ( 2 ) ...
Page 604
... infinity , then { f ( T ) } - 1 exists and has domain D ( T " ) . PROOF . We let g ( 2 ) = ( 2 - x ) " f ( 2 ) , x e o ( T ) . Then g and 1 / g are in F ( T ) , and ƒ ( T ) = Ang ( T ) . Thus f ( T ) X = D ( T " ) , and [ f ( T ) -1 [ g ...
... infinity , then { f ( T ) } - 1 exists and has domain D ( T " ) . PROOF . We let g ( 2 ) = ( 2 - x ) " f ( 2 ) , x e o ( T ) . Then g and 1 / g are in F ( T ) , and ƒ ( T ) = Ang ( T ) . Thus f ( T ) X = D ( T " ) , and [ f ( T ) -1 [ g ...
Page 641
... infinity we assigned a bounded operator f ( 4 ) such that mapping ƒ → ƒ ( A ) was a homomorphism . In this section we suppose A is the infinitesimal generator of a strongly continuous group of operators T ( t ) , ∞ < t < ∞ , and show ...
... infinity we assigned a bounded operator f ( 4 ) such that mapping ƒ → ƒ ( A ) was a homomorphism . In this section we suppose A is the infinitesimal generator of a strongly continuous group of operators T ( t ) , ∞ < t < ∞ , and show ...
Contents
Preliminary Concepts | 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 ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element ergodic exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f 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 S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ