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 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 o ( 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 o ( x ) , o ( x1 ) ≤ o and o ( y ) , o ( y1 ) ...
Page 2218
... PROOF . In view of Lemma 23 , the proof may be restricted to the case where the Boolean algebra B is complete . Let { E } be a weakly convergent generalized sequence in B and suppose that its limit E is a projection . It must be shown ...
... PROOF . In view of Lemma 23 , the proof may be restricted to the case where the Boolean algebra B is complete . Let { E } be a weakly convergent generalized sequence in B and suppose that its limit E is a projection . It must be shown ...
Page 2281
... proof is similar to that of Lemma 17 and will be omitted . We turn now to the representation of E * X * when E * € * and has finite uniform multiplicity n . The results are parallel to those in X. The weaker form of completeness enjoyed ...
... proof is similar to that of Lemma 17 and will be omitted . We turn now to the representation of E * X * when E * € * and has finite uniform multiplicity n . The results are parallel to those in X. The weaker form of completeness enjoyed ...
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