## Linear Operators: Spectral theory |

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Page 970

In this notation the formula in Theorem 11 becomes ( vf ) ( m ) = SR [ x , m ] f ( x )

dx , TeLi ( R ) , me Mo In Theorem 9 the domain of t was extended to L2 ( R ) ; we

wish to obtain a

In this notation the formula in Theorem 11 becomes ( vf ) ( m ) = SR [ x , m ] f ( x )

dx , TeLi ( R ) , me Mo In Theorem 9 the domain of t was extended to L2 ( R ) ; we

wish to obtain a

**similar**integral representation for this extension . Let E be the ...Page 1379

The proof for is exactly

be as in Theorem 18 . Then an isolated point to e Mo ( T ) is an isolated

singularity of 0.1 ; ( or , equivalently , of 0jj ) . Moreover , Pis ( { 20 } ) is the

residue at ho ...

The proof for is exactly

**similar**. Q.E.D. 30 COROLLARY . Let T , T , 1 , 0 ;, 0 , etc. ,be as in Theorem 18 . Then an isolated point to e Mo ( T ) is an isolated

singularity of 0.1 ; ( or , equivalently , of 0jj ) . Moreover , Pis ( { 20 } ) is the

residue at ho ...

Page 1383

... it follows from Corollary 29 that the entire interval -00 < < o belongs to the

resolvent set of T .. The reader will have no difficulty in carrying out an exactly

iii ) and ...

... it follows from Corollary 29 that the entire interval -00 < < o belongs to the

resolvent set of T .. The reader will have no difficulty in carrying out an exactly

**similar**computation and deriving an exactly**similar**result in case ( ii ) . In cases (iii ) and ...

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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