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 ( \ ) | ~ | A ( λ ) | as | λ | → ∞o . 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 ( \ ) | ~ | A ( λ ) | as | λ | → ∞o . 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 ) . Σ 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 ) . Σ PROOF . By the ...
Contents
SPECTRAL OPERATORS | 1924 |
Introduction | 1927 |
Relations Between a Spectral Operator and Its Scalar | 1950 |
Copyright | |
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A₁ adjoint operator algebra of projections Amer arbitrary B*-algebra B₁ Boolean algebra Borel sets boundary conditions bounded linear operator bounded operator closed operator Colojoară commuting compact complex numbers complex plane contains converges Corollary countably additive Definition dense differential operator disjoint Doklady Akad E-measurable eigenvalues elements equation equivalent exists Foias follows from Theorem formal differential operator formula function f H₁ H₂ Hence Hilbert space hypothesis identity inequality integral invariant inverse L₁ Lebesgue Lemma Math matrix multiplicity norm operators in Hilbert perturbation polynomial PROOF proved quasi-nilpotent resolution restriction Russian S₁ satisfies scalar operator scalar type operator scalar type spectral Section semi-group sequence shows spectral measure spectral operator spectral theory spectrum strong operator topology subset subspace sufficiently type spectral operator unbounded unique vector weakly complete zero