## Linear Operators: Spectral theory |

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Page 1110

We saw in the course of proving

and by

case ...

We saw in the course of proving

**Lemma**16 that the function det(I-I-T) of that**lemma**is continuous in T(cf. the remark which follows**Lemma**16). Thus, by (b)and by

**Lemma**11, to prove (f) in general we have only to prove (f) in the specialcase ...

Page 1226

Part (a) follows immediately from

from part (a) and

symmetric operator with dense domain has a unique minimal closed symmetric

extension.

Part (a) follows immediately from

**Lemma**5(b), and part (b) follows immediatelyfrom part (a) and

**Lemma**5(c). Q.E.D. It follows from**Lemma**6(b) that anysymmetric operator with dense domain has a unique minimal closed symmetric

extension.

Page 1696

By

carrier which is a compact subset C, of I, ... it follows from

3.12 that there is a unique extension G of F to a distribution in D,(D) such that the

...

By

**Lemma**14 there is a sequence {F,} of elements of D(I), each of which has acarrier which is a compact subset C, of I, ... it follows from

**Lemmas**13, 3.43 and3.12 that there is a unique extension G of F to a distribution in D,(D) such that the

...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

48 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 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