## Linear Operators: Spectral theory |

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Page 1144

is contained in an angular sector of opening less than a p . Let N > O be an

integer , and

...

**Suppose**that each of the s regions into which the plane is divided by these arcsis contained in an angular sector of opening less than a p . Let N > O be an

integer , and

**suppose**that the resolvent of T satisfies the inequality R ( 2 ; T ) = 0...

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= ulx | ? , 2 e ( T ) , so that if \ x is bounded , ( Tx , x ) is bounded below .

Conversely ,

Theorem ...

**Suppose**that such a u exists . Then , by Theorem XII . 2 . 6 , ( Tx , x ) = E ( dx ) x2= ulx | ? , 2 e ( T ) , so that if \ x is bounded , ( Tx , x ) is bounded below .

Conversely ,

**suppose**that for each n , en = ( - 0 , - n ) n o ( T ) is non - void . ByTheorem ...

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( 18 ) In the interval [ 0 , 0 ) ,

t ) ) 2 ( b ) 1700 19 ( t ) / 3 < oo , Co ( q ( + ) ) * Modt < oo , Jm Ig ( t ) / 5 / 2 for large

M . Then the essential spectrum of 1 is empty ( Wintner [ 8 ] ) . . ( 19 ) In the ...

( 18 ) In the interval [ 0 , 0 ) ,

**suppose**that ( a ) lim q ( t ) = - 00 , t + 00 lim sup ( g ' (t ) ) 2 ( b ) 1700 19 ( t ) / 3 < oo , Co ( q ( + ) ) * Modt < oo , Jm Ig ( t ) / 5 / 2 for large

M . Then the essential spectrum of 1 is empty ( Wintner [ 8 ] ) . . ( 19 ) In the ...

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

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