Linear Operators, Part 2 |
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Page 2169
... Borel functions , note that for a fixed continuous function g the set of all bounded Borel functions ƒ for which ( vi ) ( fg ) ( T ) = ƒ ( T ) g ( T ) , includes all continuous functions . Furthermore , if the equation ( vi ) holds for ...
... Borel functions , note that for a fixed continuous function g the set of all bounded Borel functions ƒ for which ( vi ) ( fg ) ( T ) = ƒ ( T ) g ( T ) , includes all continuous functions . Furthermore , if the equation ( vi ) holds for ...
Page 2188
... Borel measurable functions g for which √ ,, 9 ( μ ) E1 ( dμ ) = √ g ( f ( x ) ) E ( dA ) f ( A ) is clearly linear and closed in the set of all bounded Borel functions . Since this set contains every characteristic function of a Borel ...
... Borel measurable functions g for which √ ,, 9 ( μ ) E1 ( dμ ) = √ g ( f ( x ) ) E ( dA ) f ( A ) is clearly linear and closed in the set of all bounded Borel functions . Since this set contains every characteristic function of a Borel ...
Page 2233
... Borel sets whose closures are in U , by the equation Qox = f ( TE ( e ) X ) x , x = E ( e ) X . Now , using the machinery established in Lemma 6 , ƒ ( T ) may be defined as follows . - 8 DEFINITION . Let T be a spectral operator with ...
... Borel sets whose closures are in U , by the equation Qox = f ( TE ( e ) X ) x , x = E ( e ) X . Now , using the machinery established in Lemma 6 , ƒ ( T ) may be defined as follows . - 8 DEFINITION . Let T be a spectral operator with ...
Contents
SPECTRAL OPERATORS | 1924 |
The Canonical Reduction of a Spectral Operator | 1939 |
Bounded Spectral Operators in Hilbert Space | 1947 |
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 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