## Linear Operators: Spectral theory |

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Page 1469

Thus

( b ) then follows immediately from the preceding lemma . Q . E . D . Next we turn

...

Thus

**statement**( a ) follows immediately from the preceding lemma . To prove**statement**( b ) , note that r is finite below ho , but not below any a > 20 .**Statement**( b ) then follows immediately from the preceding lemma . Q . E . D . Next we turn

...

Page 1653

20 and Fin H ( + 1 ) ( I ) , ( cf . Definition 15 ( i ) ) . To prove ( ii ) and the final

i ) .

**Statement**( iii ) follows from**statement**( ii ) and the fact that Flu + 1 2 Flu ) for all k20 and Fin H ( + 1 ) ( I ) , ( cf . Definition 15 ( i ) ) . To prove ( ii ) and the final

**statement**of the lemma , we first note that it is evident for k 2 0 from Definition 15 (i ) .

Page 1756

Hence we find that if yl < r , f ( y ) = 0 , and

uniqueness of the function V of the theorem is an evident consequence of

existence ...

Hence we find that if yl < r , f ( y ) = 0 , and

**statement**( i ) is fully proved . ( B ) Theuniqueness of the function V of the theorem is an evident consequence of

**statement**( i ) . Moreover ,**statement**( i ) enables us to reduce the proof of theexistence ...

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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