Linear Operators: Spectral operators |
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Page 2232
... preceding theorem it follows that E ( e ) D ( Q ) ≤ D ( Q ) and QE ( e ) x = E ( e ) Qx for every x in D ( Q ) and every e in Σ . PROOF . Let { e } be as in the preceding proof , and let x be in D ( Q ) . Then lim , E ( e ) E ( e „ ) x ...
... preceding theorem it follows that E ( e ) D ( Q ) ≤ D ( Q ) and QE ( e ) x = E ( e ) Qx for every x in D ( Q ) and every e in Σ . PROOF . Let { e } be as in the preceding proof , and let x be in D ( Q ) . Then lim , E ( e ) E ( e „ ) x ...
Page 2396
... preceding lemma , by Lemma 1 , and by formulas ( 2a ) and ( 2b ) , | B ( λ ) | ~ | A ( A ) ] as [ A ] → ∞ . The present corollary now follows from the preceding lemma and from Corollary 6 along the same lines of proof as Corollary 9 ...
... preceding lemma , by Lemma 1 , and by formulas ( 2a ) and ( 2b ) , | B ( λ ) | ~ | A ( A ) ] as [ A ] → ∞ . The present corollary now follows from the preceding lemma and from Corollary 6 along the same lines of proof as Corollary 9 ...
Page 2455
... preceding lemma be H2 , H1 , H1 , we obtain the present corollary . Q.E.D. = 5 COROLLARY . Under the hypotheses of the preceding corollary , U ( H1 , H2 ) is an isometric mapping of Σ ( H1 , H2 ) onto Σ ( H2 , H1 ) . PROOF . By the ...
... preceding lemma be H2 , H1 , H1 , we obtain the present corollary . Q.E.D. = 5 COROLLARY . Under the hypotheses of the preceding corollary , U ( H1 , H2 ) is an isometric mapping of Σ ( H1 , H2 ) onto Σ ( H2 , H1 ) . PROOF . By the ...
Contents
SPECTRAL OPERATORS | 1924 |
Spectral Operators | 1925 |
Terminology and Preliminary Notions | 1928 |
Copyright | |
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A₁ algebra Amer analytic applications arbitrary B-space Banach Banach space Boolean algebra Borel sets boundary bounded Chapter clear closed commuting compact complex constant contains continuous converges Corollary corresponding defined Definition denote dense determined differential operator Doklady Akad elements equation equivalent established example exists extension finite follows formula function given gives H₁ Hence Hilbert space hypothesis identity integral invariant inverse Lemma limit linear operators Math multiplicity Nauk SSSR norm normal perturbation plane positive preceding present problem Proc projections PROOF properties prove range resolution resolvent restriction Russian satisfies scalar type sequence shown shows similar spectral measure spectral operator spectrum subset subspace sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector zero