Linear Operators: Spectral operators |
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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 ) = ƒ ( 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 ) = ƒ ( T ) g ( T ) , includes all continuous functions . Furthermore , if the equation ( vi ) ...
Page 2188
... 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 8 in the complex ...
... 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 8 in the complex ...
Page 2262
... bounded pointwise convergent sequence { f } , then it follows from ( 4 ) that it holds . for the limit function f = lim ƒn · Hence ( 5 ) holds if ƒ is any bounded Borel function vanishing at λ = 0 and at λ = vō1 . A repetition of this ...
... bounded pointwise convergent sequence { f } , then it follows from ( 4 ) that it holds . for the limit function f = lim ƒn · Hence ( 5 ) holds if ƒ is any bounded Borel function vanishing at λ = 0 and at λ = vō1 . A repetition of this ...
Contents
SPECTRAL OPERATORS | 1924 |
Introduction | 1927 |
Relations Between a Spectral Operator and Its Scalar | 1950 |
Copyright | |
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A₁ adjoint operator algebra of projections Amer arbitrary B*-algebra B₁ Boolean algebra Borel sets boundary conditions bounded linear operator bounded operator closed operator Colojoară commuting compact complex numbers complex plane contains converges Corollary countably additive Definition dense differential operator disjoint Doklady Akad E-measurable eigenvalues elements equation equivalent exists Foias follows from Theorem formal differential operator formula function f H₁ H₂ Hence Hilbert space hypothesis identity inequality integral invariant inverse L₁ Lebesgue Lemma Math matrix multiplicity norm operators in Hilbert perturbation polynomial PROOF proved quasi-nilpotent resolution restriction Russian S₁ satisfies scalar operator scalar type operator scalar type spectral Section semi-group sequence shows spectral measure spectral operator spectral theory spectrum strong operator topology subset subspace sufficiently type spectral operator unbounded unique vector weakly complete zero