## Linear Operators: Spectral theory |

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Results 1-3 of 83

Page 984

The set of functions f in L ( R ) for which s

dense in Ly ( R ) . Proof . It follows from Lemma 3 . 6 that the set of all functions in

L2 ( R , B , u ) which

The set of functions f in L ( R ) for which s

**vanishes**in a neighborhood of infinity isdense in Ly ( R ) . Proof . It follows from Lemma 3 . 6 that the set of all functions in

L2 ( R , B , u ) which

**vanish**outside of compact sets is dense in this space , and ...Page 997

Let o be a bounded measurable function on R . Then a point m , in Ř is in the

complement of the spectral set of q if and only if there are neighborhoods V of the

identity in R and U of m , such that the transform t ( pf )

in ...

Let o be a bounded measurable function on R . Then a point m , in Ř is in the

complement of the spectral set of q if and only if there are neighborhoods V of the

identity in R and U of m , such that the transform t ( pf )

**vanishes**on U for every fin ...

Page 1650

subsets of I and let F be in D ( I ) . If F

PROOF . The proofs of the first four parts of this lemma are left to the reader as an

exercise . To prove ( v ) , we must show from our hypothesis that F ( 0 ) = 0 if q is ...

subsets of I and let F be in D ( I ) . If F

**vanishes**in each set Iq , it**vanishes**in UqlaPROOF . The proofs of the first four parts of this lemma are left to the reader as an

exercise . To prove ( v ) , we must show from our hypothesis that F ( 0 ) = 0 if q is ...

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### Contents

IX | 859 |

extensive presentation of applications of the spectral theorem | 911 |

Miscellaneous Applications | 937 |

Copyright | |

20 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 |

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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 give 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