## Linear Operators: Self Adjoint Operators in Hilbert Space. Spectral theory. Part II |

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Page 860

A B -

B -

+ y * , ( xy ) * = y * x * ( ax ) * = ax * , ( x * ) * = x . All of the examples mentioned ...

A B -

**algebra**X is commutative in case xy yx for all x and y in X. An involution in aB -

**algebra**X is a mapping x + x * of X into itself with the properties ( x + y ) * = **+ y * , ( xy ) * = y * x * ( ax ) * = ax * , ( x * ) * = x . All of the examples mentioned ...

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2 LEMMA . The

in which the operation * of involution is defined by equation ( i ) is a B * -

Our chief objective in this section is to characterize commutative B * -algebras .

2 LEMMA . The

**algebra**B ( H ) of all bounded linear operators in Hilbert space yin which the operation * of involution is defined by equation ( i ) is a B * -

**algebra**.Our chief objective in this section is to characterize commutative B * -algebras .

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be based upon two closely related commutative algebras of operators in the

Hilbert space L2 ( R ) . One of these algebras , namely the

preceding section , we have met before . For convenience , its definition and

some of its ...

be based upon two closely related commutative algebras of operators in the

Hilbert space L2 ( R ) . One of these algebras , namely the

**algebra**A of thepreceding section , we have met before . For convenience , its definition and

some of its ...

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### Contents

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Preliminary Notions | 865 |

Copyright | |

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additive Akad algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure multiplicity Nauk neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero