Linear Operators: Spectral operators |
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Page 2188
... set Д and let g be a bounded Borel measurable function defined on the complex plane . Then √g ( f ( \ ) ) E ( dλ ) = Sun 9 ( μ ) E ( ƒ ̃ 1 ( dμ ) ) , ƒЄ EB ( A , E ) . - 1 ( A ) PROOF . Let f be in EB ( A , E ) , and for every Borel set ...
... set Д and let g be a bounded Borel measurable function defined on the complex plane . Then √g ( f ( \ ) ) E ( dλ ) = Sun 9 ( μ ) E ( ƒ ̃ 1 ( dμ ) ) , ƒЄ EB ( A , E ) . - 1 ( 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 ( ƒ ) ...
... 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 ( ƒ ) ...
Page 2233
... Borel sets whose closures are in U , by the equation e Q。x = ƒ ( T | E ( e ) X ) x , x = E ( e ) X . Now , using ... 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 e Q。x = ƒ ( T | E ( e ) X ) x , x = E ( e ) X . Now , using ... 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 |
Spectral Operators | 1925 |
Terminology and Preliminary Notions | 1928 |
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 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