## Linear Operators: Spectral theory |

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Page 868

If I is a closed ideal in the commutative B - algebra X then the quotient algebra X /

? is isometrically isomorphic to the field of

maximal . PROOF . If I is not maximal it is properly contained in an ideal and so X

/ I ...

If I is a closed ideal in the commutative B - algebra X then the quotient algebra X /

? is isometrically isomorphic to the field of

**complex**numbers if and only if I ismaximal . PROOF . If I is not maximal it is properly contained in an ideal and so X

/ I ...

Page 872

and each x in X define x ( a ) = lim Pn ( a ) where { P . } is a ... For a fixed do e G

the map x + x ( 20 ) is a homomorphism of X into the field of

**complex**variable that { P . ( 2 ) } also converges uniformly on G . For each 2 in Gand each x in X define x ( a ) = lim Pn ( a ) where { P . } is a ... For a fixed do e G

the map x + x ( 20 ) is a homomorphism of X into the field of

**complex**numbers .Page 1156

We leave it to the reader to show that the character group of this group is

algebraically and topologically isomorphic with the additive group of real

numbers modulo 27 , or , equivalently , with the multiplicative group of

numbers of unit ...

We leave it to the reader to show that the character group of this group is

algebraically and topologically isomorphic with the additive group of real

numbers modulo 27 , or , equivalently , with the multiplicative group of

**complex**numbers of unit ...

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### Contents

IX | 859 |

extensive presentation of applications of the spectral theorem | 911 |

Miscellaneous Applications | 937 |

Copyright | |

20 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 give 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