## Linear Operators: Spectral theory |

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Page 1455

(a) If T is a closed symmetric operator in Hilbert space which is bounded below

and whose essential spectrum o,(T) does not intersect the interval (– od, 2) of the

real axis, we say that T is

(a) If T is a closed symmetric operator in Hilbert space which is bounded below

and whose essential spectrum o,(T) does not intersect the interval (– od, 2) of the

real axis, we say that T is

**finite**below A. (b) If t is a formally symmetric formal ...Page 1459

Q.E.D. 30 COROLLARY. A formally positive formally symmetric formal differential

operator r is

bounded below. Thus the present corollary follows from Corollary 7 and

Definition ...

Q.E.D. 30 COROLLARY. A formally positive formally symmetric formal differential

operator r is

**finite**below zero. PROOF. It is obvious from Definition 20 that t isbounded below. Thus the present corollary follows from Corollary 7 and

Definition ...

Page 1460

Then, if t is

HT1 is

generality that A = 0. By Corollary 24(b), Corollary XII.4.18, and Corollary 26, To(r

) has a ...

Then, if t is

**finite**below A, and the leading coefficient of t+1, never vanishes, t-HT1 is

**finite**below A. PRoof. It is clear that we may suppose without loss ofgenerality that A = 0. By Corollary 24(b), Corollary XII.4.18, and Corollary 26, To(r

) has a ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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### 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