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 2169
... bounded Borel functions , note that for a fixed continuous function g the set of all bounded Borel functions ƒ for which ( vi ) ( fg ) ( T ) = f ( T ) g ( T ) , includes all continuous functions . Furthermore , if the equation ( vi ) ...
... bounded Borel functions , note that for a fixed continuous function g the set of all bounded Borel functions ƒ for which ( vi ) ( fg ) ( T ) = f ( T ) g ( T ) , includes all continuous functions . Furthermore , if the equation ( vi ) ...
Page 2233
... bounded Borel sets whose closures are in U , by the equation Qox = ƒ ( T | E ( e ) X ) x , x = E ( e ) X . Now , using the machinery established in Lemma 6 , f ( T ) may be defined as follows . = 8 DEFINITION . Let T be a spectral ...
... bounded Borel sets whose closures are in U , by the equation Qox = ƒ ( T | E ( e ) X ) x , x = E ( e ) X . Now , using the machinery established in Lemma 6 , f ( T ) may be defined as follows . = 8 DEFINITION . Let T be a spectral ...
Page 2262
... bounded Borel functions vanishing at λ = 0 and at λovo1 . Since it is obvious from ( 4 ) that T ( ƒ ) = 0 if ƒ is any Borel function vanish- ing except at λ = 0 and at λ = vő1 , it follows that ( 5 ) is valid for every pair of bounded Borel ...
... bounded Borel functions vanishing at λ = 0 and at λovo1 . Since it is obvious from ( 4 ) that T ( ƒ ) = 0 if ƒ is any Borel function vanish- ing except at λ = 0 and at λ = vő1 , it follows that ( 5 ) is valid for every pair of bounded Borel ...
Contents
SPECTRAL OPERATORS | 1924 |
The Resolvent of a Spectral Operator | 1935 |
An Operational Calculus for Bounded Spectral | 1941 |
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