Linear Operators: Spectral operators |
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Page 2064
... f , which is the well known uniqueness theorem for the Fourier transform . Both of these statements are contained in 1 THEOREM . For f in L1 the function f is continuous on and ƒ ( ∞ ) = 0. Also f = 0 only if ƒ ( s ) = 0 for every s in S.
... f , which is the well known uniqueness theorem for the Fourier transform . Both of these statements are contained in 1 THEOREM . For f in L1 the function f is continuous on and ƒ ( ∞ ) = 0. Also f = 0 only if ƒ ( s ) = 0 for every s in S.
Page 2189
... ( f ) S ( g ) if ƒ and g are characteristic functions of sets in 2. For a fixed characteristic function f the set of g in EB ( A , E ) for which ( ii ) holds is linear ; and , in view of ( i ) , it is closed . Thus , since it contains all ...
... ( f ) S ( g ) if ƒ and g are characteristic functions of sets in 2. For a fixed characteristic function f the set of g in EB ( A , E ) for which ( ii ) holds is linear ; and , in view of ( i ) , it is closed . Thus , since it contains all ...
Page 2262
... { 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 argument shows that ( 5 ) still holds if ...
... { 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 argument shows that ( 5 ) still holds if ...
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