Linear Operators: Spectral operators |
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Page 1953
... PROOF . First suppose that 0. If TX is not dense then , by Corollary II.3.13 , there is an x * in X * with x * 0 and ... PROOF . The proof will be divided into two cases depending on whether the projection E ( { 0 } ) = 0 or not . First ...
... PROOF . First suppose that 0. If TX is not dense then , by Corollary II.3.13 , there is an x * in X * with x * 0 and ... PROOF . The proof will be divided into two cases depending on whether the projection E ( { 0 } ) = 0 or not . First ...
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 ...
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