## 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 ...

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 ...

Page 1454

Q . E . D . 23 LEMMA . If T is a closed symmetric operator in Hilbert space , and T

is bounded below , then ( a ) the essential spectrum of T is a subset of the real

axis which is bounded below ; ( b ) the deficiency

.

Q . E . D . 23 LEMMA . If T is a closed symmetric operator in Hilbert space , and T

is bounded below , then ( a ) the essential spectrum of T is a subset of the real

axis which is bounded below ; ( b ) the deficiency

**indices**of T are equal . PROOF.

Page 1611

( 1 ) If the essential spectrum of t 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 ( 2

...

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

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

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

...

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### Contents

IX | 859 |

extensive presentation of applications of the spectral theorem | 911 |

Miscellaneous Applications | 937 |

Copyright | |

20 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 adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f give given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero