## Linear Operators: Spectral theory |

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Page 898

If we put E(0) = 0 when Ó n g(T) is void, then

Theorem 1 and

spectral measure associated, in

If we put E(0) = 0 when Ó n g(T) is void, then

**Corollary**4 follows immediately fromTheorem 1 and

**Corollary**IX.3.15. Q.E.D. 5 DEFINITION. The uniquely definedspectral measure associated, in

**Corollary**4, with the normal operator T is called ...Page 1301

However, as vo satisfies an equation of order 2n, vo must be identically zero. This

contradiction completes the proof. Q.E.D. 23

differential operator of order n on an interval I with end points a, b, and suppose

that ...

However, as vo satisfies an equation of order 2n, vo must be identically zero. This

contradiction completes the proof. Q.E.D. 23

**CoRollARY**. Let t be a formaldifferential operator of order n on an interval I with end points a, b, and suppose

that ...

Page 1459

Q.E.D. 30

operator r is finite below zero. PROOF. It is obvious from Definition 20 that t is

bounded below. Thus the present

Definition ...

Q.E.D. 30

**COROLLARY**. A formally positive formally symmetric formal differentialoperator r is finite below zero. PROOF. It is obvious from Definition 20 that t is

bounded below. Thus the present

**corollary**follows from**Corollary**7 andDefinition ...

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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