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 2149
... dense in the complex plane . PROOF . If the resolvent set is dense , then any two analytic , or even continuous , extensions of R ( λ ; T ) x must coincide on their common domain of continuity . Q.E.D. All of the special type operators ...
... dense in the complex plane . PROOF . If the resolvent set is dense , then any two analytic , or even continuous , extensions of R ( λ ; T ) x must coincide on their common domain of continuity . Q.E.D. All of the special type operators ...
Page 2156
... dense in X. Since M2 is dense in X , the manifold ( λ1I − T ) TM M2 + { x | ( λ1I − T ) x = 0 } is dense in X , so that - ( λ1I — T ) 1 ( λ2 I — T ) ˇX + { x | ( λ2I − T ) x = 0 } + { x | ( ^ 21 - T ) x = 0 } is also dense in X. By ...
... dense in X. Since M2 is dense in X , the manifold ( λ1I − T ) TM M2 + { x | ( λ1I − T ) x = 0 } is dense in X , so that - ( λ1I — T ) 1 ( λ2 I — T ) ˇX + { x | ( λ2I − T ) x = 0 } + { x | ( ^ 21 - T ) x = 0 } is also dense in X. By ...
Page 2159
... dense in T 。. PROOF . It is clear that the union of intervals of constancy is open . To see that it is dense , let y be a closed subarc of To having positive length and let 0 Yn = { λo | λo € y , | λ — λ 。| " | R ( λ ; T ) | ≤ 1 , do ...
... dense in T 。. PROOF . It is clear that the union of intervals of constancy is open . To see that it is dense , let y be a closed subarc of To having positive length and let 0 Yn = { λo | λo € y , | λ — λ 。| " | R ( λ ; T ) | ≤ 1 , do ...
Contents
SPECTRAL OPERATORS | 1924 |
An Operational Calculus for Bounded Spectral | 1941 |
Bounded Spectral Operators in Hilbert Space | 1947 |
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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