## Linear Operators: Spectral theory |

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Page 1190

Q . E . D . Most of the considerations in this chapter and the next will be directed

towards an

following definition . 7 DEFINITION . The

, y ) ...

Q . E . D . Most of the considerations in this chapter and the next will be directed

towards an

**operator**which is either**symmetric**or self adjoint according to thefollowing definition . 7 DEFINITION . The

**operator**T is said to be**symmetric**if ( Tx, y ) ...

Page 1223

In the theory of bounded

for if T is everywhere defined and

the situation is quite different . Consider , as an example , an

In the theory of bounded

**operators**, we have only to verify**symmetry**( T * 2 T ) ,for if T is everywhere defined and

**symmetric**, then T * = T . But if T is unboundedthe situation is quite different . Consider , as an example , an

**operator**which will ...Page 1272

Maximal

then it has proper symmetric extensions provided both of its deficiency indices

are different from zero . A maximal

proper ...

Maximal

**symmetric operators**. 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 . A maximal

**symmetric operator**is one which has noproper ...

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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