## Linear Operators: Spectral theory |

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Results 1-3 of 76

Page 1215

-(e).. Thus ss f(s) W,(s, 2),(ds) exists in the mean square sense and equals (Uf),(2

), proving (c). Q.E.D. Using the notation of the

that, by Lemma 9, s". (UI),(2)W.C. 2),(A) = E(-n, n)P(t)a → F(T)a = U.*F = f,.

-(e).. Thus ss f(s) W,(s, 2),(ds) exists in the mean square sense and equals (Uf),(2

), proving (c). Q.E.D. Using the notation of the

**preceding**proof we let F = (Uf), sothat, by Lemma 9, s". (UI),(2)W.C. 2),(A) = E(-n, n)P(t)a → F(T)a = U.*F = f,.

Page 1378

Moreover, in the course of the proof

was shown that if Åo is any point in A, there exists a small open subinterval N of A

, containing Ao, such that the set of restrictions 61, ..., 6, of ol, ..., or to IXN is a ...

Moreover, in the course of the proof

**preceding**the statement of Theorem 23, itwas shown that if Åo is any point in A, there exists a small open subinterval N of A

, containing Ao, such that the set of restrictions 61, ..., 6, of ol, ..., or to IXN is a ...

Page 1419

To prove the corollary it suffices to make the change of variable t → —t in the

24 is not negative for t sufficiently close to zero, then it is bounded, and Theorem

23 ...

To prove the corollary it suffices to make the change of variable t → —t in the

**preceding**corollary. Q.E.D. PRoof of THEOREM 24. If the function q of Theorem24 is not negative for t sufficiently close to zero, then it is bounded, and Theorem

23 ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

48 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

additive adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero