## Linear Operators: Spectral theory |

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Page 878

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

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

definition . 12 DEFINITION . Let x be an element of a commutative B * - algebra

and let fe C ...

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

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

definition . 12 DEFINITION . Let x be an element of a commutative B * - algebra

and let fe C ...

Page 942

Thus every eigenfunction of T , which

finite dimensional continuous function . Hence N is orthogonal to every

eigenfunction of T , except to those

Theorem X . 3 .

Thus every eigenfunction of T , which

**corresponds**to a non - zero eigenvalue is afinite dimensional continuous function . Hence N is orthogonal to every

eigenfunction of T , except to those

**corresponding**to 2 = 0 . It follows fromTheorem X . 3 .

Page 962

Now each f in Li ( R )

vanishes at infinity , and we always have tfl . ... then operating on a function f in

L2 ( R ) by the projection E ( e )

function ...

Now each f in Li ( R )

**corresponds**to some continuous function tf on M whichvanishes at infinity , and we always have tfl . ... then operating on a function f in

L2 ( R ) by the projection E ( e )

**corresponds**to multiplying the**corresponding**function ...

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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