## Linear Operators: Spectral Theory : Self Adjoint Operators in Hilbert Space, Volume 2 |

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Page 2257

3 that E ( o ; T ) = E ( 70 ) ; R ) for each

clear that as o runs over the family K of all

runs over the family of all

3 that E ( o ; T ) = E ( 70 ) ; R ) for each

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

**compact**open subsets of o ( T ) , 7 ( 0 )runs over the family of all

**compact**open subsets of o ( R ) which do not contain ...Page 2360

It will also be shown that T - v is

, it will follow that B ( u ) = R ( u ; T + P ) is

large , so that the theorem will be proved . Let u be in V ; . To show that IT ' ' R ( u

...

It will also be shown that T - v is

**compact**. From this , ( iii ) , and Theorem VI . 5 . 4, it will follow that B ( u ) = R ( u ; T + P ) is

**compact**for u in V , and i sufficientlylarge , so that the theorem will be proved . Let u be in V ; . To show that IT ' ' R ( u

...

Page 2462

The operator C is

is complete . Q . E . D . 12 LEMMA . If C is a

uniformly bounded sequence of operators in H converging strongly to zero ...

The operator C is

**compact**by Corollary V1 . 5 . 5 , and thus proof of Corollary 11is complete . Q . E . D . 12 LEMMA . If C is a

**compact**operator in H , and { Tn } is auniformly bounded sequence of operators in H converging strongly to zero ...

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