## Linear Operators: Spectral theory |

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Page 1152

The existence of an invariant measure on a group satisfying the second axiom of

countability was first shown by Haar [ 1 ] , and the question of uniqueness was

first discussed by ... Points in R have

The existence of an invariant measure on a group satisfying the second axiom of

countability was first shown by Haar [ 1 ] , and the question of uniqueness was

first discussed by ... Points in R have

**positive measure**if and only if R is discrete .Page 1276

and that this sequence can be represented in the form ' u ( dt ) , 820 where u is a

, 820 , [ + 00 converge . Is the

and that this sequence can be represented in the form ' u ( dt ) , 820 where u is a

**positive**Borel**measure**on the real axis such that all the integrals to 1012 ; u ( dt ), 820 , [ + 00 converge . Is the

**measure**u unique ? This fundamental problem of ...Page 1341

If { uisj } is an n x n

uis } ) is a Hilbert space . The proof of this theorem will be based on the following

lemma . 11 LEMMA . Let { uis } be a

...

If { uisj } is an n x n

**positive**matrix**measure**defined on the real line , then L2 ( {uis } ) is a Hilbert space . The proof of this theorem will be based on the following

lemma . 11 LEMMA . Let { uis } be a

**positive**matrix**measure**whose elements are...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

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 neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result Russian satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero