## Linear Operators: Spectral theory |

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Page 1051

Statements ( ii ) and ( iii ) are

formulae Sen 9 ( x ) dx = Sem P ( Ux ) dx , Sen g ( xx ) dx = ( 2xl - srn9 ( x ) dx ,

which are valid for every Lebesgue integrable function q . Statement ( iv ) will

follow in ...

Statements ( ii ) and ( iii ) are

**evident**consequences of Definition 1 and of theformulae Sen 9 ( x ) dx = Sem P ( Ux ) dx , Sen g ( xx ) dx = ( 2xl - srn9 ( x ) dx ,

which are valid for every Lebesgue integrable function q . Statement ( iv ) will

follow in ...

Page 1695

Then it is

number of the terms of this series fail to vanish in any compact subset of I , it is

Then it is

**evident**that the series * ( x ) = 4 ; ( « ) j = 1 converges to a ... only a finitenumber of the terms of this series fail to vanish in any compact subset of I , it is

**evident**that m QE CO° ( I ) . q = lim EniQ , m + 00 j = 1 Thus , by Definitions 3 .Page 1756

( B ) The uniqueness of the function V of the theorem is an

of statement ( i ) . Moreover , statement ( i ) enables us to reduce the proof of the

existence of the function V to the proof of the following statement . ( ii ) For each r

...

( B ) The uniqueness of the function V of the theorem is an

**evident**consequenceof statement ( i ) . Moreover , statement ( i ) enables us to reduce the proof of the

existence of the function V to the proof of the following statement . ( ii ) For each r

...

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### Contents

BAlgebras | 859 |

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 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 satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero