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 2299
... sufficiently large and for C1 , we have - | R ( μ ; T + P ) — R ( μ ; T ) | = | B ( μ ) — R ( μ ; T ) | since for sufficiently large n , ∞ 1 00 -m ≤ 2M dñ1 Σ ( | ^ n ] + d „ ) mv d „ TM ( 2M | A | ) m m = 1 ≤ 8M2 | A | d2 ( | λn ] + ...
... sufficiently large and for C1 , we have - | R ( μ ; T + P ) — R ( μ ; T ) | = | B ( μ ) — R ( μ ; T ) | since for sufficiently large n , ∞ 1 00 -m ≤ 2M dñ1 Σ ( | ^ n ] + d „ ) mv d „ TM ( 2M | A | ) m m = 1 ≤ 8M2 | A | d2 ( | λn ] + ...
Page 2360
... sufficiently large . From this it will then follow as above that the function ƒ ( μ ) : Ru ; TP ) f is uniformly ... sufficiently large , so that the theorem will be proved . i . Let μ be in V1 . To show that | TR ( μ ; T ) A ...
... sufficiently large . From this it will then follow as above that the function ƒ ( μ ) : Ru ; TP ) f is uniformly ... sufficiently large , so that the theorem will be proved . i . Let μ be in V1 . To show that | TR ( μ ; T ) A ...
Page 2394
... sufficiently small μ Є P + , such that σ3 and os are continuous in t and μ for 0 ≤t < ∞ and μ sufficiently small , and such that σ3 ( t , μ ) ~ e - itu ; σ'ś ( t , μ ) ~ —iμe - itu ; as too , for all μe P + such that I ( p ) 0 , | μ ...
... sufficiently small μ Є P + , such that σ3 and os are continuous in t and μ for 0 ≤t < ∞ and μ sufficiently small , and such that σ3 ( t , μ ) ~ e - itu ; σ'ś ( t , μ ) ~ —iμe - itu ; as too , for all μe P + such that I ( p ) 0 , | μ ...
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