## Linear Operators: Spectral theory |

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Page 875

It will be

commutative B * - algebra X into the algebra C ( 1 ) of all continuous functions on

the structure space 1 of X is an isometric isomorphism of X onto all of C ( 1 ) .

It will be

**shown**that the homomorphism x + x ( • ) ( see Theorem 2 . 9 ) of acommutative B * - algebra X into the algebra C ( 1 ) of all continuous functions on

the structure space 1 of X is an isometric isomorphism of X onto all of C ( 1 ) .

Page 981

If H ( T ( f ) ) does not vanish identically for f in Li ( R ) then , as was

first part of the proof of Theorem 3 . 11 , there is a continuous character h on R

with H ( T ( 1 ) ) = Jah ( x ) f ( x ) dx , feL ( R ) . The converse part of Theorem 3 .

If H ( T ( f ) ) does not vanish identically for f in Li ( R ) then , as was

**shown**in thefirst part of the proof of Theorem 3 . 11 , there is a continuous character h on R

with H ( T ( 1 ) ) = Jah ( x ) f ( x ) dx , feL ( R ) . The converse part of Theorem 3 .

Page 1161

That spectral synthesis is not possible for all functions in Lo was

Schwartz [ 2 ] for Euclidean space of three dimensions . It has recently been

on the ...

That spectral synthesis is not possible for all functions in Lo was

**shown**by L .Schwartz [ 2 ] for Euclidean space of three dimensions . It has recently been

**shown**by M . Paul Malliavin that spectral synthesis is not possible for all functionson the ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 869 |

Commutative BAlgebras | 877 |

Copyright | |

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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 remark 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