Linear Operators: Spectral operators |
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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 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 2232
... proof , Qx = = lim QoE ( en ) x = lim QE ( en ) x . Since is closed , it follows that x is in D ( Q ) and that Qx QQ ... PROOF . Let { e } be as in the preceding proof , and let x be in D ( Q ) . Then lim , E ( e ) E ( e , ) x = E ( e ) ...
... proof , Qx = = lim QoE ( en ) x = lim QE ( en ) x . Since is closed , it follows that x is in D ( Q ) and that Qx QQ ... PROOF . Let { e } be as in the preceding proof , and let x be in D ( Q ) . Then lim , E ( e ) E ( e , ) x = E ( e ) ...
Page 2396
... proof , ∞ 2 | ƒ4 ( 0 ) | ≤ \ r \ ̄1 { ƒ ~ _ ~ \ q ( 0 ) | - ds + 0 t -2iu ( st ) , e 13q ( 8 ) ds . It follows from this formula just as in the proof of Lemma 1 ( cf. the para- graph following formula ( 14 ) ) that lim 0011711 ueP + f ...
... proof , ∞ 2 | ƒ4 ( 0 ) | ≤ \ r \ ̄1 { ƒ ~ _ ~ \ q ( 0 ) | - ds + 0 t -2iu ( st ) , e 13q ( 8 ) ds . It follows from this formula just as in the proof of Lemma 1 ( cf. the para- graph following formula ( 14 ) ) that lim 0011711 ueP + f ...
Contents
SPECTRAL OPERATORS | 1924 |
14 | 1983 |
Sufficient Conditions | 2134 |
Copyright | |
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A₁ adjoint operator algebra of projections Amer analytic arbitrary asymptotic B₁ Banach space Boolean algebra Borel set boundary conditions bounded Borel function bounded linear operator bounded operator commuting compact complete Boolean algebra complex numbers complex plane continuous functions converges Corollary countably additive Definition denote differential operator Doklady Akad domain E₁ eigenvalues elements equation exists finite number follows from Lemma follows from Theorem follows immediately formal differential operator formula function f H₁ H₂ Hence Hilbert space hypothesis identity inequality inverse L₁ Lebesgue Math multiplicity Nauk SSSR norm operators in Hilbert perturbation PROOF properties prove quasi-nilpotent resolution Russian satisfies scalar type operator scalar type spectral Section sequence shows spectral measure spectral operator spectral theory spectrum subset Suppose trace class type spectral operator unbounded uniformly bounded vector zero