## Linear Operators: Spectral theory |

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Page 1142

The validity of the

from its validity in the range 2 Spoo and from Lemma 9.14 . Q.E.D. In what follows

, we will use the symbols p and n to denote the continuous extension to the

classes ...

The validity of the

**present**theorem in the range 1 < p = 2 now follows at oncefrom its validity in the range 2 Spoo and from Lemma 9.14 . Q.E.D. In what follows

, we will use the symbols p and n to denote the continuous extension to the

classes ...

Page 1684

By Corollary 2 and Hölder's inequality , each ( k - m ) th derivative of any lth

derivative of F belongs to L , ( EK ) ( and has compact carrier ) for 1 Sm . Hence ,

it is quite sufficient to prove the

Corollary ...

By Corollary 2 and Hölder's inequality , each ( k - m ) th derivative of any lth

derivative of F belongs to L , ( EK ) ( and has compact carrier ) for 1 Sm . Hence ,

it is quite sufficient to prove the

**present**lemma for the special case m = 0. ByCorollary ...

Page 1692

Using this remark , the proofs of Corollary 6 and Lemma 8 may immediately be

extended to the

COROLLARY . Let C denote either a bounded domain in En whose boundary is

a smooth ...

Using this remark , the proofs of Corollary 6 and Lemma 8 may immediately be

extended to the

**present**case . Details are left to the reader . Q.E.D. 10COROLLARY . Let C denote either a bounded domain in En whose boundary is

a smooth ...

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### Contents

BAlgebras | 861 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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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 complete Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense derivatives determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure 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