Linear Operators: Spectral operators |
From inside the book
Results 1-3 of 89
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 |
Introduction | 1927 |
Relations Between a Spectral Operator and Its Scalar | 1950 |
Copyright | |
22 other sections not shown
Other editions - View all
Common terms and phrases
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