## Linear Operators: Spectral theory |

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Page 950

Instead of restricting our consideration to the case of the additive group of real

numbers , we shall discuss the case of a locally

denote by R . We assume throughout that R is o -

Instead of restricting our consideration to the case of the additive group of real

numbers , we shall discuss the case of a locally

**compact**Abelian group which wedenote by R . We assume throughout that R is o -

**compact**, i . e . , the union of ...Page 1150

ence of Haar measure on a locally

remarked in the text , the development presented in this section is valid for a

general non - discrete locally

there are ...

ence of Haar measure on a locally

**compact**, o -**compact**Abelian group . Asremarked in the text , the development presented in this section is valid for a

general non - discrete locally

**compact**, o -**compact**Abelian group . However ,there are ...

Page 1331

12 shows immediately that Irl , K ( t , s ; 2 ) / 2 dsdt < c . To complete the proof it is

therefore sufficient to show that every integral operator in LZ ( I ) defined by a

kernel K with | | K | | 2 = | | , | K ( t , 8 ) / 2 dsdt < 0 is

case ...

12 shows immediately that Irl , K ( t , s ; 2 ) / 2 dsdt < c . To complete the proof it is

therefore sufficient to show that every integral operator in LZ ( I ) defined by a

kernel K with | | K | | 2 = | | , | K ( t , 8 ) / 2 dsdt < 0 is

**compact**. This is a specialcase ...

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

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