## Linear Operators: Spectral theory |

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Page 1092

By Lemma 5 and Corollary 4 , and the elementary fact that any compact operator

may be approximated in norm by a sequence of operators Tn with

finitedimensional

T has finite ...

By Lemma 5 and Corollary 4 , and the elementary fact that any compact operator

may be approximated in norm by a sequence of operators Tn with

finitedimensional

**range**, it is enough to prove the lemma in the special case thatT has finite ...

Page 1134

Then , retracing the steps of the above argument , we can conclude that ( I - E )

TE , = 0 for each 2 in C. Hence T leaves the

, and the set F of projections Ex , de C , subdiagonalizes T. To prove the second ...

Then , retracing the steps of the above argument , we can conclude that ( I - E )

TE , = 0 for each 2 in C. Hence T leaves the

**range**of each projection E , invariant, and the set F of projections Ex , de C , subdiagonalizes T. To prove the second ...

Page 1395

Then ( E ( Q ) U ) x = ( 1 - E ( { 0 } ) ( 11 –T ) ) x = ( 11 —T ) x which shows that the

V of a which is disjoint from 01 , and let f ( u ) = ( 2 - u - 1 if u € V and f ( u ) = 0 if u

...

Then ( E ( Q ) U ) x = ( 1 - E ( { 0 } ) ( 11 –T ) ) x = ( 11 —T ) x which shows that the

**range**of the projection E ( 01 ) contains the**range**of T. Choose a neighborhoodV of a which is disjoint from 01 , and let f ( u ) = ( 2 - u - 1 if u € V and f ( u ) = 0 if u

...

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