Linear Operators: Spectral operators |
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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 ( A ; 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 ( A ; T ) x must coincide on their common domain of continuity . Q.E.D. All of the special type operators ...
Page 2156
... dense in X , the manifold ( λ21 − T ) TM M2 + { x | ( λ1I − T ) ˇx = 0 } is dense in X , so that - ( λ1I — T ) Ŋ ( λ2 I — T ) 2X + { x | ( ^ 1I − T ) 1x = 0 } - + { x | ( \ 2 1 − T ) 1x = 0 } - = is also dense in X. By Lemma 7 , σ ...
... dense in X , the manifold ( λ21 − T ) TM M2 + { x | ( λ1I − T ) ˇx = 0 } is dense in X , so that - ( λ1I — T ) Ŋ ( λ2 I — T ) 2X + { x | ( ^ 1I − T ) 1x = 0 } - + { x | ( \ 2 1 − T ) 1x = 0 } - = is also dense in X. By Lemma 7 , σ ...
Page 2159
... dense on à 。. Some results in this direction will be found in the next four lemmas . 11 LEMMA ( G ) . The union of all intervals of constancy relative to T is an open set dense in T 。. PROOF . It is clear that the union of intervals ...
... dense on à 。. Some results in this direction will be found in the next four lemmas . 11 LEMMA ( G ) . The union of all intervals of constancy relative to T is an open set dense in T 。. PROOF . It is clear that the union of intervals ...
Contents
SPECTRAL OPERATORS | 1924 |
Spectral Operators | 1925 |
Terminology and Preliminary Notions | 1928 |
Copyright | |
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A₁ algebra Amer analytic applications arbitrary B-space Banach Banach space Boolean algebra Borel sets boundary bounded Chapter clear closed commuting compact complex constant contains continuous converges Corollary corresponding defined Definition denote dense determined differential operator Doklady Akad elements equation equivalent established example exists extension finite follows formula function given gives H₁ Hence Hilbert space hypothesis identity integral invariant inverse Lemma limit linear operators Math multiplicity Nauk SSSR norm normal perturbation plane positive preceding present problem Proc projections PROOF properties prove range resolution resolvent restriction Russian satisfies scalar type sequence shown shows similar spectral measure spectral operator spectrum subset subspace sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector zero