## Linear Operators: Spectral theory |

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Page 890

... then the function E is a resolution of the identity for T and the operational calculus is given by the

... then the function E is a resolution of the identity for T and the operational calculus is given by the

**formula**( vi ) f ( T ) = Sop , ( 2 ) E ( da ) . where the integral is defined as the finite sum L- | ( 2 ; ) E ( 2 ) .Page 1089

The basic properties of the characteristic numbers un ( T ) are stated in the following lemma and corollaries . be nat 2 LEMMA . The characteristic numbers un ( T ) of a compact operator are given by the following

The basic properties of the characteristic numbers un ( T ) are stated in the following lemma and corollaries . be nat 2 LEMMA . The characteristic numbers un ( T ) of a compact operator are given by the following

**formula**: ( 11 L'est ...Page 1363

basis for this

basis for this

**formula**is found in Theorem XII.2.10 which asserts that the projection in the resolution of the identity for T corresponding to ( 1g , 22 ) may be calculated from the resolvent by the**formula**1 E ( ( 17 , 13 ) ) = lim lim ...### What people are saying - Write a review

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### Contents

BAlgebras | 859 |

Miscellaneous Applications | 937 |

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 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