## Linear operators. 2. Spectral theory : self adjoint operators in Hilbert Space |

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Page 1152

The existence of an invariant

The existence of an invariant

**measure**on a group satisfying the second axiom of countability was first shown by Haar [ 1 ] ... Other results concerning**measures**invariant under transformations are found in Oxtoby and Ulam [ 1 ] .Page 1153

Since the

Since the

**measure**space ( R , E , 2 ) is a o - finite**measure**space the theory of integration as developed in Chapter III may be used as a basis for the theory developed in Sections 3. - 4 . In particular we should notice that the ...Page 1154

o - compact group R and let 2 be a Haar

o - compact group R and let 2 be a Haar

**measure**in R. Then the product**measure**à xà is a Haar**measure**in Rx R. PROOF . Since the product group R ( 2 ) = RxR is locally compact and o - compact , it has a Haar**measure**2 ( 2 ) defined on ...### What people are saying - Write a review

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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 2: Spectral Theory, Self Adjoint Operators in Hilbert ... Nelson Dunford,Jacob T. Schwartz No preview available - 1988 |

Linear Operators, Part 2: Spectral Theory, Self Adjoint Operators in Hilbert ... Nelson Dunford,Jacob T. Schwartz No preview available - 1988 |

Linear Operators, Part 2: Spectral Theory, Self Adjoint Operators in Hilbert ... Nelson Dunford,Jacob T. Schwartz No preview available - 1988 |

### Common terms and phrases

additive adjoint operator 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 satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero