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 ( λ1I − T ) TM M2 + { x | ( λ2I − T ) 1x = 0 } is dense in X , so that - ( λ1I — T ) 1 ( λ2 I — T ) 2X + { x | ( λ1I − T ) 2x = 0 } + { x ( λ2I - T ) Nx = 0 } is also dense in X. By Lemma 7 , o ( x ) ≤ y if ...
... dense in X , the manifold ( λ1I − T ) TM M2 + { x | ( λ2I − T ) 1x = 0 } is dense in X , so that - ( λ1I — T ) 1 ( λ2 I — T ) 2X + { x | ( λ1I − T ) 2x = 0 } + { x ( λ2I - T ) Nx = 0 } is also dense in X. By Lemma 7 , o ( x ) ≤ y if ...
Page 2159
... dense on г. 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 г. PROOF . It is clear that the union of intervals of ...
... dense on г. 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 г. PROOF . It is clear that the union of intervals of ...
Contents
SPECTRAL OPERATORS | 1924 |
Spectral Operators | 1925 |
Terminology and Preliminary Notions | 1928 |
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 Borel function 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 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