## Linear Operators: Spectral operators |

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Page 2299

Hence we find that for n

( u ; T + P ) = Blu ) . Since Blu ) is clearly the product of the compact operator R ( u

; T ) and a bounded operator , it follows that T + P is a discrete operator .

Hence we find that for n

**sufficiently**large , each u in Cn is in p ( T + P ) and that R( u ; T + P ) = Blu ) . Since Blu ) is clearly the product of the compact operator R ( u

; T ) and a bounded operator , it follows that T + P is a discrete operator .

Page 2300

Since E , + Xiai Elan ; T ) = 1 , 1 - Ê has a finite dimensional range for all p . Thus ,

by Lemma VII . 6 . 7 , I - E , has finite dimensional range for all

Since E is a countably additive spectral resolution , we have Elu ; T + P ) ( I – E ...

Since E , + Xiai Elan ; T ) = 1 , 1 - Ê has a finite dimensional range for all p . Thus ,

by Lemma VII . 6 . 7 , I - E , has finite dimensional range for all

**sufficiently**large p .Since E is a countably additive spectral resolution , we have Elu ; T + P ) ( I – E ...

Page 2394

Then there exists a solution oz ( t , u ) of the equation to = u o , defined for 0 St <

oo and for all

and Me for 0 St < oo and je

...

Then there exists a solution oz ( t , u ) of the equation to = u o , defined for 0 St <

oo and for all

**sufficiently**small u € P + , such that og and o ' s are continuous in tand Me for 0 St < oo and je

**sufficiently**small , and such that ozlt , p ) ~ e - itu ; oślt...

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### Contents

SPECTRAL OPERATORS | 1924 |

An Operational Calculus for Bounded Spectral | 1941 |

Part | 1950 |

Copyright | |

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