## 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 /

3 is isometrically isomorphic to the field of

maximal . Proof . If I is not maximal it is properly contained in an ideal and so X / F

...

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

3 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 / F

...

Page 872

and each x in X define x ( 2 ) = lim Pm ( 2 ) where { Pn } is a ... For a fixed 20 € G

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

**complex**variable that { Pn ( 2 ) } also converges uniformly on G. For each â in Gand each x in X define x ( 2 ) = lim Pm ( 2 ) where { Pn } is a ... For a fixed 20 € G

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

**complex**numbers .Page 1157

Then a

function g which is analytic in a neighborhood of t and is such that g ( x ) = f ( z )

for all z in this neighborhood for which 121 1 . Making use of this theorem and an

...

Then a

**complex**number t of modulus 1 is outside o ( ! ) if and only if there exists afunction g which is analytic in a neighborhood of t and is such that g ( x ) = f ( z )

for all z in this neighborhood for which 121 1 . Making use of this theorem and an

...

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### Contents

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Copyright | |

57 other sections not shown

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additive Akad algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure multiplicity Nauk neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero