Linear Operators: General theory |
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Page 557
... PROOF . This follows from Theorem 3. We have seen that T always satisfies a non - zero polynomial equation P ( T ) = 0 , where the roots of P ( 2 ) are the spectral points of T. Q.E.D. Theorem 3 enables us to define the operator f ( T ) ...
... PROOF . This follows from Theorem 3. We have seen that T always satisfies a non - zero polynomial equation P ( T ) = 0 , where the roots of P ( 2 ) are the spectral points of T. Q.E.D. Theorem 3 enables us to define the operator f ( T ) ...
Page 576
... follows from Theorem 20 that σ = σ ( T ) = 4. Q.E.D. 22 THEOREM . Let 21 , . . . , λ be poles of T , let v1 , orders , and let σ = { 21 , ... , λ % } . Then , for f in F ( T ) , f ( T ) E ( 0 ) = PROOF . If k vi − 1 f ( m ) ( 2¿ ) ΣΣ i ...
... follows from Theorem 20 that σ = σ ( T ) = 4. Q.E.D. 22 THEOREM . Let 21 , . . . , λ be poles of T , let v1 , orders , and let σ = { 21 , ... , λ % } . Then , for f in F ( T ) , f ( T ) E ( 0 ) = PROOF . If k vi − 1 f ( m ) ( 2¿ ) ΣΣ i ...
Page 689
... follows from Corollary II.3.13 . Q.E.D. = = COROLLARY . In a reflexive space , the averages A ( a ) are strongly convergent if they are bounded and if T ( n ) / n converges to zero strongly . PROOF . This follows from Theorem 1 and Theorem ...
... follows from Corollary II.3.13 . Q.E.D. = = COROLLARY . In a reflexive space , the averages A ( a ) are strongly convergent if they are bounded and if T ( n ) / n converges to zero strongly . PROOF . This follows from Theorem 1 and Theorem ...
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