## Linear Operators: Spectral theory |

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Page 984

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

is 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 Ly ( R ) for which s

**vanishes**in a neighborhood of infinityis dense 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 993

It remains to be proved that the number ay is independent of the open set V . If f is

in Lj ( R ) L2 ( R ) , f

open subset V , of V , then the above proof shows that ( 0f ) ( m ) = Qy for every ...

It remains to be proved that the number ay is independent of the open set V . If f is

in Lj ( R ) L2 ( R ) , f

**vanishes**on the complement of V , and f ( m ) = 1 for m in anopen subset V , of V , then the above proof shows that ( 0f ) ( m ) = Qy for every ...

Page 997

Let y 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 mo such that the transform t ( of )

in ...

Let y 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 mo such that the transform t ( of )

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

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

15 other sections not shown

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additive adjoint operator 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 neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result Russian satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero