## Linear Operators: Spectral theory |

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Page 950

... i.e., the union of countably many compact sets. Every such group has a non-

negative countably additive

on compact sets, positive or infinite on open sets, invariant under translation, ...

... i.e., the union of countably many compact sets. Every such group has a non-

negative countably additive

**measure**which is defined on the Borel sets 2, finiteon compact sets, positive or infinite on open sets, invariant under translation, ...

Page 1152

The existence of an invariant

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

concerning

Ulam [1].

The existence of an invariant

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

concerning

**measures**invariant under transformations are found in Oxtoby andUlam [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 R is a ...

Since the

**measure**space (R, 2, A) is a g-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 R is a ...

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### Contents

SPECTRAL THEORY | 858 |

868 | 885 |

Miscellaneous Applications | 937 |

Copyright | |

38 other sections not shown

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