Linear Operators: Part III: Spectral Operators [by] Nelson Dunford and Jacob T. Schwartz, with the Assistance of William G. Bade and Robert G. Bartle, Volume 1 |
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Page 1956
... follows from Theorem 2 that À is in the point spectrum of T. If E ( { } ) = 0 then it follows from Theorem 2 that IT is one - to - one . By Corollary 7.12 the set ( ÀI — T ) X is dense in X and hence is in the continuous spectrum of T ...
... follows from Theorem 2 that À is in the point spectrum of T. If E ( { } ) = 0 then it follows from Theorem 2 that IT is one - to - one . By Corollary 7.12 the set ( ÀI — T ) X is dense in X and hence is in the continuous spectrum of T ...
Page 2246
... follows that R ( A ) is a bounded operator whose range is contained in the domain of C. It is clear then that ( AIC ) R ( λ ) x = x_for x in H and R ( A ) ( AIC ) x = x for x in D ( C ) , so that R ( X ) = R ( λ ; C ) and λ σ ( C ) . On ...
... follows that R ( A ) is a bounded operator whose range is contained in the domain of C. It is clear then that ( AIC ) R ( λ ) x = x_for x in H and R ( A ) ( AIC ) x = x for x in D ( C ) , so that R ( X ) = R ( λ ; C ) and λ σ ( C ) . On ...
Page 2459
... follows at once . If x , € Σac ( H ) and lim ̧ → ∞ x = x , then , by what we have already proved , we may write xy1 + y2 + ys , where y1 € Σac ( H ) and y2 , Y3 are orthogonal to Σac ( H ) . But , since x , € Σac ( H ) we have ( x , y ) ...
... follows at once . If x , € Σac ( H ) and lim ̧ → ∞ x = x , then , by what we have already proved , we may write xy1 + y2 + ys , where y1 € Σac ( H ) and y2 , Y3 are orthogonal to Σac ( H ) . But , since x , € Σac ( H ) we have ( x , y ) ...
Contents
SPECTRAL OPERATORS | 1924 |
The Resolvent of a Spectral Operator | 1935 |
An Operational Calculus for Bounded Spectral | 1941 |
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A₁ adjoint operator algebra of projections Amer analytic arbitrary B-algebra B*-algebra B₁ Banach space Boolean algebra Borel sets boundary conditions bounded linear operator bounded operator closed operator commuting compact complex numbers complex plane converges Corollary countably additive Definition denote dense differential operator Doklady Akad domain E₁ eigenvalues elements equation exists finite number follows from Lemma follows from Theorem formal differential operator formula function f H₁ H₂ Hence Hilbert space hypothesis identity inequality integral invariant inverse L₁ Lebesgue Lemma Math multiplicity Nauk SSSR norm operators in Hilbert perturbation polynomial PROOF properties prove quasi-nilpotent resolution Russian S₁ satisfies scalar type operator scalar type spectral Section sequence shows spectral measure spectral operator spectral theory spectrum subset subspace Suppose trace class type spectral operator unbounded uniformly bounded unique vector zero