## Linear Operators: Spectral theory |

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Page 1239

Conversely, let T be a self adjoint

restriction of To to a subspace Q3 of Q(T*) determined by a symmetric family of

linearly independent boundary conditions B, (a) = 0, i = 1,..., k, and we have only

to ...

Conversely, let T be a self adjoint

**extension**of T. Then by Lemma 26, T, is therestriction of To to a subspace Q3 of Q(T*) determined by a symmetric family of

linearly independent boundary conditions B, (a) = 0, i = 1,..., k, and we have only

to ...

Page 1261

23 If an operator T has a closed linear

linear

is called the closure of T. (a) There exists a densely defined operator with no

closed ...

23 If an operator T has a closed linear

**extension**there exists a unique closedlinear

**extension**T such that if T is any closed linear**extension**of T then TC T1. Tis called the closure of T. (a) There exists a densely defined operator with no

closed ...

Page 1397

Q.E.D. It follows from Theorem 5 and Corollary 4 that the set of nonisolated points

of the spectrum of a self adjoint

particular

...

Q.E.D. It follows from Theorem 5 and Corollary 4 that the set of nonisolated points

of the spectrum of a self adjoint

**extension**T of To(r) is independent of theparticular

**extension**chosen, i.e., is independent of the particular set of boundary...

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

34 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 Akad algebra Amer analytic applied assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complete 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 measure multiplicity neighborhood norm obtained partial positive preceding present problem projection PRoof properties prove range regular remark representation respectively restriction result satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero