## Linear Operators: Spectral operators |

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Page 2299

Hence we find that for n

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

u ; T ' ) and a bounded operator , it follows that I + P is a discrete operator .

Hence we find that for n

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

u ; T ' ) and a bounded operator , it follows that I + P is a discrete operator .

Page 2300

Since Ep + Erzi Ean ; T ) = 1 , 1 – É , has a finite dimensional range for all p . Thus

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

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

Since Ep + Erzi Ean ; T ) = 1 , 1 – É , has a finite dimensional range for all p . Thus

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

**sufficiently**largep . Since E is a countably additive spectral resolution , we have E ( u ; T + P ) ( I ...

Page 2360

It will be shown below that IT ' R ( u ; T ) A S } for u in V ; and i

From this it will then follow as above that the function f ( u ) = R ( u ; T + P ) f is

uniformly bounded . It will also be shown that T - v is compact . From this , ( iii ) ,

and ...

It will be shown below that IT ' R ( u ; T ) A S } for u in V ; and i

**sufficiently**large .From this it will then follow as above that the function f ( u ) = R ( u ; T + P ) f is

uniformly bounded . It will also be shown that T - v is compact . From this , ( iii ) ,

and ...

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