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 2174
... Hence we have 1 2 | ett ( S + T ) | = | ettsettт | ≤ettsettr≤ M1M2 for all tЄ R. Hence if this boundedness implied that S + T was a spectral operator , we would have a contradiction to McCarthy's [ 2 , I ] modification of Kakutani's ...
... Hence we have 1 2 | ett ( S + T ) | = | ettsettт | ≤ettsettr≤ M1M2 for all tЄ R. Hence if this boundedness implied that S + T was a spectral operator , we would have a contradiction to McCarthy's [ 2 , I ] modification of Kakutani's ...
Page 2295
... Hence fu ( T ) | E ( o ; T ) X is the inverse of ( I − T ) | E ( o ; T ) X , which shows that o ( T | E ( 0 ; T ) X ) ≤ 0 . - = Suppose next that E ( σ ; T ) = 0 , but that o is not void . Then E ( λ ; T ) E ( Ao ; T ) = E ( λ ; T ) E ...
... Hence fu ( T ) | E ( o ; T ) X is the inverse of ( I − T ) | E ( o ; T ) X , which shows that o ( T | E ( 0 ; T ) X ) ≤ 0 . - = Suppose next that E ( σ ; T ) = 0 , but that o is not void . Then E ( λ ; T ) E ( Ao ; T ) = E ( λ ; T ) E ...
Page 2357
... Hence ( P + N ) ( SMI ) = P ( SMI ) ~ ' + N ( S – XI ) - " = P ( T − XI ) - L + N ( S − XI ) - = is a bounded operator which is compact if P ( TM ) is compact ( cf. VI.5.4 ) . In all cases ( a ) , ( b ) , ( c ) of the theorem , we may ...
... Hence ( P + N ) ( SMI ) = P ( SMI ) ~ ' + N ( S – XI ) - " = P ( T − XI ) - L + N ( S − XI ) - = is a bounded operator which is compact if P ( TM ) is compact ( cf. VI.5.4 ) . In all cases ( a ) , ( b ) , ( c ) of the theorem , we may ...
Contents
SPECTRAL OPERATORS | 1924 |
The Resolvent of a Spectral Operator | 1935 |
An Operational Calculus for Bounded Spectral | 1941 |
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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