## Linear Operators: Spectral theory |

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Page 1272

If T is a symmetric operator with dense domain , then it has proper symmetric

extensions provided both of its

maximal symmetric operator is one which has no proper symmetric extensions ;

hence , a ...

If T is a symmetric operator with dense domain , then it has proper symmetric

extensions provided both of its

**deficiency indices**are different from zero . Amaximal symmetric operator is one which has no proper symmetric extensions ;

hence , a ...

Page 1398

Therefore T , has a proper symmetric extension T2 , and the proof is complete . Q

. E . D . 8 COROLLARY . Let t be a formally self adjoint formal differential operator

defined on an interval 1 . If the minimum of the

Therefore T , has a proper symmetric extension T2 , and the proof is complete . Q

. E . D . 8 COROLLARY . Let t be a formally self adjoint formal differential operator

defined on an interval 1 . If the minimum of the

**deficiency indices**of To ( t ) is k ...Page 1610

( 16 ) Suppose that [ a , b ) = [ 0 , 00 ) , that the

and that there exists a sequence { In } of square - integrable functions such that in

vanishes in the interval [ 0 , n ] , \ tal = 1 , and l ( at ) in SK . Then the interval [ 2 - K

...

( 16 ) Suppose that [ a , b ) = [ 0 , 00 ) , that the

**deficiency indices**of t are equaland that there exists a sequence { In } of square - integrable functions such that in

vanishes in the interval [ 0 , n ] , \ tal = 1 , and l ( at ) in SK . Then the interval [ 2 - K

...

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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