## Linear Operators: Spectral theory |

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Page 1000

If { { n } were known to be uniformly convergent in a neighborhood of U , the

analyticity of its limit to would be

sequence in is uniformly convergent on any region containing an interval of the

real axis ...

If { { n } were known to be uniformly convergent in a neighborhood of U , the

analyticity of its limit to would be

**clear**. Unfortunately it is not**clear**that thesequence in is uniformly convergent on any region containing an interval of the

real axis ...

Page 1308

This contradiction proves our assertion . It is

work for any values of i and j , 1 si , j < 2 . It then follows readily that ( after suitable

normalization of C , C2 , Dy , and D2 ) we may write ( ts , g ) - ( 1 , tg ) = C ( 1 ) C2

...

This contradiction proves our assertion . It is

**clear**that a similar argument willwork for any values of i and j , 1 si , j < 2 . It then follows readily that ( after suitable

normalization of C , C2 , Dy , and D2 ) we may write ( ts , g ) - ( 1 , tg ) = C ( 1 ) C2

...

Page 1652

Then , since | Flu ) 2 | F | , for each F in H ( k ) ( I ) , it is

to some F in L ( I ) . Similarly , since | F | « 2120F \ , for each F in H ( * ) ( I ) and

each index J such that J1 < k , it is

Then , since | Flu ) 2 | F | , for each F in H ( k ) ( I ) , it is

**clear**that { Fn } convergesto some F in L ( I ) . Similarly , since | F | « 2120F \ , for each F in H ( * ) ( I ) and

each index J such that J1 < k , it is

**clear**that if J S k , the sequence ...### What people are saying - Write a review

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### Contents

IX | 859 |

extensive presentation of applications of the spectral theorem | 911 |

Miscellaneous Applications | 937 |

Copyright | |

20 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 adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f give given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero