## Linear Operators: Spectral theory |

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Page 860

... B - algebra because it lacks a

adjoined to such an algebra so that the extended algebra is a B - algebra . Let X

be an algebra satisfying all the requirements of a B - algebra except that X has no

... B - algebra because it lacks a

**unit**e . We shall show how a**unit**may beadjoined to such an algebra so that the extended algebra is a B - algebra . Let X

be an algebra satisfying all the requirements of a B - algebra except that X has no

**unit**.Page 865

The following lemma shows that the opposite inclusion holds in case X , has the

same

whose

.

The following lemma shows that the opposite inclusion holds in case X , has the

same

**unit**as X . 9 LEMMA . Let x be an element of a B - subalgebra X , of Xwhose

**unit**is the same as that of X . Then o ( x ) Çoc ( x ) while the boundary of 0.

Page 979

The algebra A is , by definition , the B - algebra obtained by adjoining the

T ( L ( R ) ) . Its elements have the form al + A where A is in T ( L ( R ) ) . This

algebra A is also a B * - algebra and for f in Ly ( R ) the Hilbert space adjoint of T (

) is ...

The algebra A is , by definition , the B - algebra obtained by adjoining the

**unit**I toT ( L ( R ) ) . Its elements have the form al + A where A is in T ( L ( R ) ) . This

algebra A is also a B * - algebra and for f in Ly ( R ) the Hilbert space adjoint of T (

) is ...

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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