Linear Operators, Part 2 |
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Page 2141
... sequences of closed sets in 2 ( T ) such that μ , ỡ , no ' , and x = lim { E ( ) x + E ( μn ) x } , x = X. Since the sequence { E ( vn ) + E ( μ ) } is strongly convergent , it is bounded ( cf. II.3.6 ) and the operators E ( vn ) + E ...
... sequences of closed sets in 2 ( T ) such that μ , ỡ , no ' , and x = lim { E ( ) x + E ( μn ) x } , x = X. Since the sequence { E ( vn ) + E ( μ ) } is strongly convergent , it is bounded ( cf. II.3.6 ) and the operators E ( vn ) + E ...
Page 2197
... sequence ( a monotone sequence ) in B. Then , if { E } is increasing , ( V Ea ) x , α lim E , x = ( \ Ea α while if { E } is decreasing , then lim Ex = ( E ) x , α α x = x , x = x . Conversely , if every monotone increasing generalized ...
... sequence ( a monotone sequence ) in B. Then , if { E } is increasing , ( V Ea ) x , α lim E , x = ( \ Ea α while if { E } is decreasing , then lim Ex = ( E ) x , α α x = x , x = x . Conversely , if every monotone increasing generalized ...
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 ...
... 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 ...
Contents
SPECTRAL OPERATORS | 1924 |
The Canonical Reduction of a Spectral Operator | 1939 |
Bounded Spectral Operators in Hilbert Space | 1947 |
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 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