## 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 deficiency

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 Ty , 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

Therefore T , has a proper symmetric extension Ty , 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 1611

( 1 ) If the essential spectrum of 1 is not the entire real axis , the deficiency

of t are equal ( 6 . 6 ) . ( 2 ) In particular , the deficiency

bounded below . ( 3 ) If for some real or complex 2 all solutions of the equation ( 4

...

( 1 ) If the essential spectrum of 1 is not the entire real axis , the deficiency

**indices**of t are equal ( 6 . 6 ) . ( 2 ) In particular , the deficiency

**indices**are equal if r isbounded below . ( 3 ) If for some real or complex 2 all solutions of the equation ( 4

...

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### Contents

IX | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Compact Groups | 945 |

Copyright | |

46 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 algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently consider constant contains continuous converges Corollary corresponding defined Definition denote dense derivatives determined 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 projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero