## Linear Operators: Spectral theory |

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Page 860

A B-

a B-

++y”, (ry)* = y” (x,v)* = &r", (r”)* = r. All of the examples mentioned above, with the

...

A B-

**algebra**3 is commutative in case a y = ya for all a and y in 3. An involution ina B-

**algebra**3 is a mapping a -> a “ of 3 into itself with the properties (a +y)* = a++y”, (ry)* = y” (x,v)* = &r", (r”)* = r. All of the examples mentioned above, with the

...

Page 868

Commutative B-Algebras In case 3 is a commutative B-

two-sided and the quotient

a B-

Commutative B-Algebras In case 3 is a commutative B-

**algebra**every ideal & istwo-sided and the quotient

**algebra**3/8 is again a commutative**algebra**. It will bea B-

**algebra**if Ø is closed (1.13). It is readily seen that every ideal & in 3 which ...Page 979

One of these algebras, namely the

met before. For convenience, its definition and some of its properties will be

restated here. For every f in L1(R) the convolution ( . )() = s.sr-y)g(y)ly, ge L.(R), ...

One of these algebras, namely the

**algebra**Şs of the preceding section, we havemet before. For convenience, its definition and some of its properties will be

restated here. For every f in L1(R) the convolution ( . )() = s.sr-y)g(y)ly, ge L.(R), ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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### 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 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