## Linear Operators: Spectral theory |

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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 | be a function in Ly ( R ) , L ( R ) whose transform of

complement of V and let A be the linear manifold in L ( R ) of elements of the form

Py ( x ) = { c [ x , m ; ] where mi e o ( Q ) + V . For each such element qy let the

map Qy ...

Let | be a function in Ly ( R ) , L ( R ) whose transform of

**vanishes**on thecomplement of V and let A be the linear manifold in L ( R ) of elements of the form

Py ( x ) = { c [ x , m ; ] where mi e o ( Q ) + V . For each such element qy let the

map Qy ...

Page 1650

Nelson Dunford, Jacob T. Schwartz, William G. Bade, Robert G. Bartle. subsets of

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

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

...

Nelson Dunford, Jacob T. Schwartz, William G. Bade, Robert G. Bartle. subsets of

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

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

...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 869 |

Commutative BAlgebras | 877 |

Copyright | |

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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 remark 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