## Linear Operators: Self Adjoint Operators in Hilbert Space. Spectral theory. Part II |

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Page 984

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

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 f

**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 ...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 ( of )

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 ( of )

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

Page 1650

If F

Lemma 2.4 , let { 91 , ... , 9p } be a finite set of functions in CO ( EN ) such that o =

1879 , and such that each function Vi

) ...

If F

**vanishes**in each set Iç , it**vanishes**in U21 & PROOF . The proofs ... UsingLemma 2.4 , let { 91 , ... , 9p } be a finite set of functions in CO ( EN ) such that o =

1879 , and such that each function Vi

**vanishes**outside some set læ . Then р F ( q) ...

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### Contents

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Preliminary Notions | 865 |

Copyright | |

61 other sections not shown

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