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 2236
... statement ( v ) remains to be proved . In the first case described in statement ( v ) , σ ( T ) is a compact subset of U , and by Corollary XV.3.5 , E ( o ( T ) ) I so that without loss of generality each of the sets e , n ≥ 1 , may be ...
... statement ( v ) remains to be proved . In the first case described in statement ( v ) , σ ( T ) is a compact subset of U , and by Corollary XV.3.5 , E ( o ( T ) ) I so that without loss of generality each of the sets e , n ≥ 1 , may be ...
Page 2239
... Statement ( d ) is obvious . Letting e e o and xe E ( e ) X , we have T ( fxe ) x = lim T ( fxe ) E ( en ) x = lim T ... Statement ( a ) clearly will follow from statement ( b ) . Statement ( b ) follows from statement ( f ) ; indeed ...
... Statement ( d ) is obvious . Letting e e o and xe E ( e ) X , we have T ( fxe ) x = lim T ( fxe ) E ( en ) x = lim T ... Statement ( a ) clearly will follow from statement ( b ) . Statement ( b ) follows from statement ( f ) ; indeed ...
Page 2476
... statement ( 76 ) holds for each ge ' such that g , ( a ) = 0 for all a Єe ,. Since ( H + V , H ) is a closed subspace of H ′ ( cf. Lemma 2 ) it follows that every ğ Є H ' such that g , ( a ) = 0 for all a € e , belongs to Σ ( H + V , H ) ...
... statement ( 76 ) holds for each ge ' such that g , ( a ) = 0 for all a Єe ,. Since ( H + V , H ) is a closed subspace of H ′ ( cf. Lemma 2 ) it follows that every ğ Є H ' such that g , ( a ) = 0 for all a € e , belongs to Σ ( H + V , H ) ...
Contents
SPECTRAL OPERATORS | 1924 |
The Resolvent of a Spectral Operator | 1935 |
An Operational Calculus for Bounded Spectral | 1941 |
Copyright | |
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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