## Linear Operators: Spectral theory |

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Page 1040

It will now be shown that yz ( 2 ) = 2N Eām ; T ) * R ( ā ; T ) * y vanishes which will

prove that y ( 2 ) is

be

) ...

It will now be shown that yz ( 2 ) = 2N Eām ; T ) * R ( ā ; T ) * y vanishes which will

prove that y ( 2 ) is

**analytic**at all the points a = 2m , so that y ( 2 ) can only fail tobe

**analytic**at the point a = 0 . To show this , note that ( 42 ( 2 ) , x ) = ( 2 ^ Elām ; T) ...

Page 1102

The determinant det ( I + zT » ) is an

z , if T , operates in finite - dimensional space , and hence more generally if T ,

has a finite - dimensional range . Thus , since a bounded convergent sequence

of ...

The determinant det ( I + zT » ) is an

**analytic**( and even a polynomial ) function ofz , if T , operates in finite - dimensional space , and hence more generally if T ,

has a finite - dimensional range . Thus , since a bounded convergent sequence

of ...

Page 1379

If 0 ( • ) is

each Borel set e with compact closure contained in 1 . Thus , by Theorem 25 , 01

, . . . , Ox is a determining set for T . Q . E . D . 28 COROLLARY . Let T , 4 , 0 ; ...

If 0 ( • ) is

**analytic**for ; > k , it follows from Theorem 18 that psi ( e ) = 0 for j > k andeach Borel set e with compact closure contained in 1 . Thus , by Theorem 25 , 01

, . . . , Ox is a determining set for T . Q . E . D . 28 COROLLARY . Let T , 4 , 0 ; ...

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### Contents

BAlgebras | 861 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

37 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

additive 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 differential equations 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 multiplicity neighborhood norm obtained partial positive preceding present problem Proc projection proof properties prove range regular remark representation respectively restriction result Russian satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero