## Linear Operators: Spectral theory |

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Page 1303

Clearly B ( 1 ) = 0 for those | which vanish in a

boundary value for ī at a . To prove the converse , let B be a boundary value at a .

Choose a function h in Co ( 1 ) which is identically equal to one in a

Clearly B ( 1 ) = 0 for those | which vanish in a

**neighborhood**of a . Thus B is aboundary value for ī at a . To prove the converse , let B be a boundary value at a .

Choose a function h in Co ( 1 ) which is identically equal to one in a

**neighborhood**...Page 1678

Let y be a second function in CO ( I ) such that y ( x ) = 1 for x in a

of Kį . Then yo - yo vanishes in a

Let y be a second function in CO ( I ) such that y ( x ) = 1 for x in a

**neighborhood**of Kį . Then yo - yo vanishes in a

**neighborhood**of Kn C ( F ) , and vanishes in a**neighborhood**of C ( F ) - K since y vanishes in the complement of K . Hence yo ...Page 1733

Q . E . D . Lemma 18 enables us to use the method of proof of Theorem 2 in the

out in the next two lemmas . 19 LEMMA . Let o be an elliptic formal partial ...

Q . E . D . Lemma 18 enables us to use the method of proof of Theorem 2 in the

**neighborhood**of the boundary of a domain with smooth boundary . This is carriedout in the next two lemmas . 19 LEMMA . Let o be an elliptic formal partial ...

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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