## Linear Operators: Spectral operators |

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Page 2169

uniform limit of analytic functions , it follows that this map is also a homomorphism

on the algebra of continuous functions . To see that it is a homomorphism on the

algebra of

uniform limit of analytic functions , it follows that this map is also a homomorphism

on the algebra of continuous functions . To see that it is a homomorphism on the

algebra of

**bounded Borel**functions , note that for a fixed continuous function g ...Page 2188

Let E be a spectral measure in the complex B - space X which is defined and

countably additive on a o - field of subsets of a set 1 and let g be a

measurable function defined on the complex plane . Then $ 9 ( F ( A ) ) E ( da ) =

S ...

Let E be a spectral measure in the complex B - space X which is defined and

countably additive on a o - field of subsets of a set 1 and let g be a

**bounded Borel**measurable function defined on the complex plane . Then $ 9 ( F ( A ) ) E ( da ) =

S ...

Page 2233

9 ) of an analytic function of a bounded operator by observing that R ( ) ; T | B ( 8 )

£ ) 3 = R ( ) ; T | F ( e ) { ) x , Ae 8 , 3 e E ( + ) { . ... ( ii ) If e is any Borel set , f ( T | E (

e ) X ) = f ( T ) | E ( e ) X . If , in particular , e is a

9 ) of an analytic function of a bounded operator by observing that R ( ) ; T | B ( 8 )

£ ) 3 = R ( ) ; T | F ( e ) { ) x , Ae 8 , 3 e E ( + ) { . ... ( ii ) If e is any Borel set , f ( T | E (

e ) X ) = f ( T ) | E ( e ) X . If , in particular , e is a

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