## Linear Operators: Spectral theory |

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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 - l exists as an**element**of Y then , since X and Y have...

Page 878

Clearly the requirement that x and g ( u ) = u be corresponding

determines the * - isomorphism uniquely and we are thus led to the following

definition . 12 DEFINITION . Let æ be an

and let fe C ...

Clearly the requirement that x and g ( u ) = u be corresponding

**elements**determines the * - isomorphism uniquely and we are thus led to the following

definition . 12 DEFINITION . Let æ be an

**element**of a commutative B * - algebraand let fe C ...

Page 1339

An

The set of all equivalence classes of

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

An

**element**F of Ly ( { ui ; } ) will be said to be a { u ish - null function if ( F ) = 0 .The set of all equivalence classes of

**elements**of Ly ( { uis } ) modulo { uis } - nullfunctions will be denoted by L ( { Wix } ) . We observe that by Lemma 7 , 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