## Linear Operators: Spectral theory |

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Page 860

A B - algebra X is commutative in case xy = yx for all x and y in X . An involution in

a

the class of analytic functions , are commutative

A B - algebra X is commutative in case xy = yx for all x and y in X . An involution in

a

**B**-**algebra**X is a mapping x + x * of X ... with the exception of Lil - 00 , 00 ) andthe class of analytic functions , are commutative

**B**-**algebras**with involutions .Page 861

given algebra X is algebraically and topologically equivalent to the

X ) . Since lx | = | vel = \ Tzel < lellTel it follows that the inverse map T - l is

continuous . To see that r is also continuous it will first be shown that t ( X ) is

closed in ...

given algebra X is algebraically and topologically equivalent to the

**B**-**algebra**T (X ) . Since lx | = | vel = \ Tzel < lellTel it follows that the inverse map T - l is

continuous . To see that r is also continuous it will first be shown that t ( X ) is

closed in ...

Page 868

Commutative

two - sided and the quotient algebra X / I is again a commutative algebra . It will

be a

Commutative

**B**-**Algebras**In case X is a commutative**B**-**algebra**every ideal I istwo - sided and the quotient algebra X / I is again a commutative algebra . It will

be a

**B**-**algebra**if I is closed ( 1 . 13 ) . It is readily seen that every ideal I in X ...### What people are saying - Write a review

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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