## Linear Operators: Spectral theory |

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Page 1152

The existence of an invariant

countability was first shown by Haar [ 1 ] , and the ... Other results concerning

The existence of an invariant

**measure**on a group satisfying the second axiom ofcountability was first shown by Haar [ 1 ] , and the ... Other results concerning

**measures**invariant under transformations are found in Oxtoby and Ulam [ 1 ] .Page 1153

Since the

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 product

group Rx ...

Since the

**measure**space ( R , E , 2 ) is a o - finite**measure**space the theory ofintegration 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 product

group Rx ...

Page 1154

( i ) o - compact group R and let à be a Haar

= Rx R is locally compact and o - compact , it has a Haar

on ...

( i ) o - compact group R and let à be a Haar

**measure**in R . Then the product**measure**2 xà is a Haar**measure**in Rx R . Proof . Since the product group R ( 2 )= Rx R is locally compact and o - compact , it has a Haar

**measure**2 ( 2 ) definedon ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 869 |

Commutative BAlgebras | 877 |

Copyright | |

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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 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 unit vanishes vector zero