Linear Operators: Part III: Spectral Operators [by] Nelson Dunford and Jacob T. Schwartz, with the Assistance of William G. Bade and Robert G. Bartle, Volume 1 |
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Page 2197
... sequence ( a monotone sequence ) in B. Then , if { E } is increasing , lim Ex = α Pa ) x , x = x , α while if { E } is decreasing , then lim Ex = ( ^ E « ) x , α α x Є X. Conversely , if every monotone increasing generalized sequence ...
... sequence ( a monotone sequence ) in B. Then , if { E } is increasing , lim Ex = α Pa ) x , x = x , α while if { E } is decreasing , then lim Ex = ( ^ E « ) x , α α x Є X. Conversely , if every monotone increasing generalized sequence ...
Page 2218
... sequence in B and suppose that its limit E is a projection . It must be shown that { E } converges strongly to E. By Lemma 6 , E is in B and so a consideration of the sequence { E - E } shows that it may be assumed that E 0. Thus , to ...
... sequence in B and suppose that its limit E is a projection . It must be shown that { E } converges strongly to E. By Lemma 6 , E is in B and so a consideration of the sequence { E - E } shows that it may be assumed that E 0. Thus , to ...
Page 2450
... sequence { n } contains no repetitions ) and that λ # 0 for n 1. By Corollary X.3.5 we have Anoo . Let r , be the distance from the eigenvalue A , to the other points on the spectrum of T , so that r , > 0 , while , of course , the sequence ...
... sequence { n } contains no repetitions ) and that λ # 0 for n 1. By Corollary X.3.5 we have Anoo . Let r , be the distance from the eigenvalue A , to the other points on the spectrum of T , so that r , > 0 , while , of course , the sequence ...
Contents
SPECTRAL OPERATORS | 1924 |
Relations Between a Spectral Operator and Its Scalar | 1950 |
The Algebras and | 1967 |
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 linear operator bounded operator closed operator commuting compact complex numbers complex plane converges Corollary countably additive Definition denote dense differential operator Doklady Akad domain E₁ 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