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 2188
... set A and let g be a bounded Borel measurable function defined on the complex plane . Then √ g ( f ( x ) ) E ( dx ) = √ , 9 ( μ ) E ( ƒ − 1 ( dμ ) ) , ƒЄ EB ( 4 , Σ ) . f ( A ) PROOF . Let f be in EB ( A , E ) , and for every Borel set ...
... set A and let g be a bounded Borel measurable function defined on the complex plane . Then √ g ( f ( x ) ) E ( dx ) = √ , 9 ( μ ) E ( ƒ − 1 ( dμ ) ) , ƒЄ EB ( 4 , Σ ) . f ( A ) PROOF . Let f be in EB ( A , E ) , and for every Borel set ...
Page 2189
... set of g in EB ( A , E ) for which ( ii ) holds is linear ; and , in view of ( i ) , it is closed . Thus , since it ... Borel set 8 in the plane , let E1 ( 8 ) = E ( ƒ − 1 ( 8 ) ) . Then , by taking g ( μ ) = μ in Lemma 8 , ( iii ) S ...
... set of g in EB ( A , E ) for which ( ii ) holds is linear ; and , in view of ( i ) , it is closed . Thus , since it ... Borel set 8 in the plane , let E1 ( 8 ) = E ( ƒ − 1 ( 8 ) ) . Then , by taking g ( μ ) = μ in Lemma 8 , ( iii ) S ...
Page 2233
... Borel sets whose closures are in U , by the equation Qox = ƒ ( T | E ( e ) X ) x , x = E ( e ) X . Now , using the ... set U such that E ( U ) I. Let { e } be an arbitrary increasing sequence of bounded Borel sets with closures contained ...
... Borel sets whose closures are in U , by the equation Qox = ƒ ( T | E ( e ) X ) x , x = E ( e ) X . Now , using the ... set U such that E ( U ) I. Let { e } be an arbitrary increasing sequence of bounded Borel sets with closures contained ...
Contents
SPECTRAL OPERATORS | 1924 |
An Operational Calculus for Bounded Spectral | 1941 |
Bounded Spectral Operators in Hilbert Space | 1947 |
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A₁ Acad adjoint operator algebra of projections Amer analytic arbitrary B-algebra B-space 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 dense differential operator Dokl Doklady Akad eigenvalues elements equation exists formal differential operator formula function f H₁ H₂ Hence Hilbert space hypothesis ibid identity inequality inverse Krein L₁ Lemma locally convex spaces multiplicity Nauk SSSR norm normal operators operators in Hilbert perturbation polynomial Proc PROOF properties prove Pure Appl quasi-nilpotent resolution Russian S₁ satisfies scalar operator scalar type operator scalar type spectral Section semi-group sequence shows spectral measure spectral operator spectral theory spectrum subset Suppose topology trace class type spectral operator unbounded uniformly bounded vector zero