## Linear Operators: Spectral theory |

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Page 1226

Part (a)

from part (a) and Lemma 5(c). Q.E.D. It follows from Lemma 6(b) that any

symmetric operator with dense domain has a unique minimal closed symmetric

extension.

Part (a)

**follows immediately**from Lemma 5(b), and part (b)**follows immediately**from part (a) and Lemma 5(c). Q.E.D. It follows from Lemma 6(b) that any

symmetric operator with dense domain has a unique minimal closed symmetric

extension.

Page 1469

It

–e/2)|f|* for f in Q(T.). Since U, e.,3 (T.) D o(To(t)), ((t–(Äo-e/2))f, f) > 0 for f in Q(To(t

)), so that t—(Åo-e/2) is formally positive, and, by Corollary 30, t is finite below ...

It

**follows immediately**from Theorems 4.1, 4.2, and XII.7.2 that (rf, f) = (T. f. f) > (20–e/2)|f|* for f in Q(T.). Since U, e.,3 (T.) D o(To(t)), ((t–(Äo-e/2))f, f) > 0 for f in Q(To(t

)), so that t—(Åo-e/2) is formally positive, and, by Corollary 30, t is finite below ...

Page 1478

Thus A, -> 00 by Corollary 26 and Corollary 27. The uniqueness of q,

distinct eigenvalues have different numbers of zeros, and since by Corollary 44

this ...

Thus A, -> 00 by Corollary 26 and Corollary 27. The uniqueness of q,

**follows****immediately**from Lemma 41. Since, by Lemma 45, eigenfunctions belonging todistinct eigenvalues have different numbers of zeros, and since by Corollary 44

this ...

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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