## Linear Operators: Spectral theory |

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Page 865

An

idempotent with 0 # lo te clearly has 0 . ( x ) Ço ( x ) . The following lemma shows

that the opposite inclusion holds in case X , has the same unit as X . 9 LEMMA .

An

**element**æ in a B - subalgebra of the form Xo = 2oXe , where e , is anidempotent with 0 # lo te clearly has 0 . ( x ) Ço ( x ) . The following lemma shows

that the opposite inclusion holds in case X , has the same unit as X . 9 LEMMA .

Page 877

Then an

Consequently the spectrum of y as an

an

Then an

**element**y in Y has an inverse in X if and only if it has an inverse in y .Consequently the spectrum of y as an

**element**of Y is the same as its spectrum asan

**element**of X . PROOF . If y - 1 exists as an**element**of Y then , since X and Y ...Page 1339

An

set of all equivalence classes of

functions will be denoted by L2 ( { Mis } ) . We observe that by Lemma 7 , the ...

An

**element**F of L ( { uis } ) will be said to be a { wiss - null function if ( F \ = 0 . Theset of all equivalence classes of

**elements**of Ly ( { uis } ) modulo { lij } - nullfunctions will be denoted by L2 ( { Mis } ) . We observe that by Lemma 7 , the ...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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