## Linear Operators: Spectral theory |

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Page 1083

of the preceding

be a positive measure space. Then an operator A in the Hilbert space L2(S, 2, u)

is of Hilbert-Schmidt class if and only if there exists a uxu measurable function ...

of the preceding

**exercise**converges in the Hilbert-Schmidt norm. 44 Let (S, 2, u)be a positive measure space. Then an operator A in the Hilbert space L2(S, 2, u)

is of Hilbert-Schmidt class if and only if there exists a uxu measurable function ...

Page 1086

Show, finally, that by choosing A(s, s) = 0 for all s in S, we obtain the result of

of

Show, finally, that by choosing A(s, s) = 0 for all s in S, we obtain the result of

**Exercise**46 as a special case of the present result. (Hint: Generalize the methodof

**Exercise**46.) 49 The operator A of Hilbert-Schmidt class is said to be of trace ...Page 1087

(Hint: For (d), use Weyl's inequality,

(Halberg) Let (S. 2, u) be a g-finite measure space. Let T, be a 1-parameter family

of bounded operators defined in a subinterval I of the parameter interval 1 < p ...

(Hint: For (d), use Weyl's inequality,

**Exercise**30.) E. Miscellaneous Earercises 50(Halberg) Let (S. 2, u) be a g-finite measure space. Let T, be a 1-parameter family

of bounded operators defined in a subinterval I of the parameter interval 1 < p ...

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

34 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

additive Akad algebra Amer analytic applied assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complete 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 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