## Linear Operators: Spectral theory |

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Page 916

The sets en will be called the

ek ) > 0 and u ( ek + 1 ) = 0 then the ordered representation is said to have

The sets en will be called the

**multiplicity**sets of the ordered representation . If u (ek ) > 0 and u ( ek + 1 ) = 0 then the ordered representation is said to have

**multiplicity**k . If ulex ) > 0 for all k , the representation is said to have infinite**multiplicity**...Page 1091

Let im ( T ) be an enumeration of the non - zero eigenvalues of T , each repeated

according to its

zero eigenvalues of Tn , with repetitions according to

...

Let im ( T ) be an enumeration of the non - zero eigenvalues of T , each repeated

according to its

**multiplicity**. Then there exist enumerations am ( Tn ) of the non -zero eigenvalues of Tn , with repetitions according to

**multiplicity**, such that m 2 1...

Page 1217

If u ( ex ) > 0 for all k , the representation is said to have infinite

ordered representations U and U of H relative to T and I respectively , with

measures u and ű , and

if u ū ...

If u ( ex ) > 0 for all k , the representation is said to have infinite

**multiplicity**. Twoordered representations U and U of H relative to T and I respectively , with

measures u and ű , and

**multiplicity**sets { en } and { ēn } will be called equivalentif u ū ...

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

15 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 adjoint operator 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