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 1953
... PROOF . The proof will be divided into two cases depending on whether the projection E ( { 0 } ) = 0 or not . First suppose that E ( { 0 } ) = 0 . Then since the range of T is closed , it follows from Corollary 12 that TX = X. By Lemma ...
... PROOF . The proof will be divided into two cases depending on whether the projection E ( { 0 } ) = 0 or not . First suppose that E ( { 0 } ) = 0 . Then since the range of T is closed , it follows from Corollary 12 that TX = X. By Lemma ...
Page 2137
... PROOF . The proof of Corollary XV.3.3 may be used to prove the present lemma . Q.E.D. 3 LEMMA ( A ) . Let o be a set of complex numbers , and o ' its com- plement . If x + y = x1 + y1 , where σ ( x ) , o ( x1 ) ≤ o and o ( y ) , o ( y1 ) ...
... PROOF . The proof of Corollary XV.3.3 may be used to prove the present lemma . Q.E.D. 3 LEMMA ( A ) . Let o be a set of complex numbers , and o ' its com- plement . If x + y = x1 + y1 , where σ ( x ) , o ( x1 ) ≤ o and o ( y ) , o ( y1 ) ...
Page 2192
... proof of the lemma . E ( 8 ) = 1 Q.E.D. 12 COROLLARY . Let A be an algebra of operators in a weakly complete B - space X. Suppose that A is topologically and algebraically isomorphic to some B - algebra of bounded continuous functions ...
... proof of the lemma . E ( 8 ) = 1 Q.E.D. 12 COROLLARY . Let A be an algebra of operators in a weakly complete B - space X. Suppose that A is topologically and algebraically isomorphic to some B - algebra of bounded continuous functions ...
Contents
SPECTRAL OPERATORS | 1924 |
An Operational Calculus for Bounded Spectral | 1941 |
Bounded Spectral Operators in Hilbert Space | 1947 |
Copyright | |
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A₁ Acad adjoint operator algebra of projections Amer analytic arbitrary B-algebra B-space 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 dense differential operator Dokl Doklady Akad eigenvalues elements equation exists formal differential operator formula function f H₁ H₂ Hence Hilbert space hypothesis ibid identity inequality inverse Krein L₁ Lemma locally convex spaces multiplicity Nauk SSSR norm normal operators operators in Hilbert perturbation polynomial Proc PROOF properties prove Pure Appl quasi-nilpotent resolution Russian S₁ satisfies scalar operator scalar type operator scalar type spectral Section semi-group sequence shows spectral measure spectral operator spectral theory spectrum subset Suppose topology trace class type spectral operator unbounded uniformly bounded vector zero