## 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

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 we denote by R. We assume throughout that R is o -**compact**, i.e. , the ...Page 1150

ence of Haar measure on a locally

ence of Haar measure on a locally

**compact**, o -**compact**Abelian group . As remarked in the text , the development presented in this section is valid for a general non - discrete locally**compact**, o -**compact**Abelian group .Page 1331

To complete the proof it is therefore sufficient to show that every integral operator in Lề ( I ) defined by a kernel K with || K || 2 = 1 $ , \ K ( t , 8 ) 2 dsdt < 0 is

To complete the proof it is therefore sufficient to show that every integral operator in Lề ( I ) defined by a kernel K with || K || 2 = 1 $ , \ K ( t , 8 ) 2 dsdt < 0 is

**compact**. This is a special case of Exercise VI.9.52 , but , for ...### What people are saying - Write a review

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### Contents

8 | 876 |

859 | 885 |

extensive presentation of applications of the spectral theorem | 911 |

Copyright | |

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additive adjoint adjoint operator algebra analytic assume basis belongs Borel set boundary conditions boundary values bounded called clear closed closure commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues elements equal equation equivalent Exercise exists extension fact finite dimensional follows follows from Lemma formal differential operator formula function given Hence Hilbert space Hilbert-Schmidt ideal identity immediately implies independent inequality integral interval invariant isometric isomorphism Lemma limit linear Ly(R mapping matrix measure multiplicity neighborhood norm obtained orthonormal positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown shows solutions spectral spectrum square-integrable statement subset subspace sufficient Suppose symmetric Theorem theory topology transform uniformly unique unit unitary vanishes vector zero