## 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

o - compact group R and let à be a Haar

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

on its ...

o - compact group R and let à be a Haar

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

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

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### Contents

IX | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Compact Groups | 945 |

Copyright | |

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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 consider constant contains continuous converges Corollary corresponding defined Definition denote dense derivatives determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function 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