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. o ' , then 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 ) ...
... PROOF . The proof of Corollary XV.3.3 may be used to prove the present lemma . Q.E.D. o ' , then 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 ) ...
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 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