## Linear Operators: Spectral operators |

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Page 2257

3 that Elo ; T ' ) = E ( + ( 0 ) ; R ) for each

is clear that as o runs over the family K of all

) runs over the family of all

3 that Elo ; T ' ) = E ( + ( 0 ) ; R ) for each

**compact**spectral set o of T . Moreover , itis clear that as o runs over the family K of all

**compact**open subsets of olT ' ) , 7 ( o) runs over the family of all

**compact**open subsets of o ( R ) which do not ...Page 2357

then it is clear that L is a bounded operator and that ( - XI ) - v = ( T – WI ) - " L .

Hence ( P + N ) ( S – XI ) - V = P ( S – XI ) - " + N ( S – XI ) - = P ( T – 2 ] ) - ' L + N (

S – AI ) - V is a bounded operator which is

cf .

then it is clear that L is a bounded operator and that ( - XI ) - v = ( T – WI ) - " L .

Hence ( P + N ) ( S – XI ) - V = P ( S – XI ) - " + N ( S – XI ) - = P ( T – 2 ] ) - ' L + N (

S – AI ) - V is a bounded operator which is

**compact**if P ( T – 21 ) - " is**compact**(cf .

Page 2462

is

. D . 12 LEMMA . If C is a

bounded ...

is

**compact**. Put C = QR1 , and D = Rg , so that V = CD . The operator C is**compact**by Corollary V1 . 5 . 5 , and thus proof of Corollary 11 is complete . Q . E. D . 12 LEMMA . If C is a

**compact**operator in H , and { Tn } is a uniformlybounded ...

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### Contents

SPECTRAL OPERATORS 1937 1941 1945 XV Spectral Operators | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

29 other sections not shown

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