## Linear Operators: Spectral theory |

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Page 993

Then it

& p is independent of V. Q.E.D. 16 THEOREM . If the bounded measurable

function y has its spectral set consisting of the single point m then , for some

complex ...

Then it

**follows**from what has just been demonstrated that av , = Qyuv , = Qy , i.e. ,& p is independent of V. Q.E.D. 16 THEOREM . If the bounded measurable

function y has its spectral set consisting of the single point m then , for some

complex ...

Page 996

Since f * q is continuous by Lemma 3.1 ( d ) it

that f * 4 +0 . From Lemma 12 ( b ) it is seen that olf * 9 ) Cole ) and from Lemma

12 ( c ) and the equation of = Tf it

of o ...

Since f * q is continuous by Lemma 3.1 ( d ) it

**follows**from the above equationthat f * 4 +0 . From Lemma 12 ( b ) it is seen that olf * 9 ) Cole ) and from Lemma

12 ( c ) and the equation of = Tf it

**follows**that o ( f * Q ) contains no interior pointof o ...

Page 1124

If E ,, E are in F and q ( En ) increases to the limit 9 ( E ) , then it

we have already proved that E is an increasing sequence of projections and E ,

SE . If Ec is the strong limit of En , then E. SE and Q ( E ) = Q ( E ) . Thus , it

...

If E ,, E are in F and q ( En ) increases to the limit 9 ( E ) , then it

**follows**from whatwe have already proved that E is an increasing sequence of projections and E ,

SE . If Ec is the strong limit of En , then E. SE and Q ( E ) = Q ( E ) . Thus , it

**follows**...

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