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

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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 bounded

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 bounded

measurable function defined on the complex plane . Then 2 ) ) E ( d ) ) = l g ( u ) E

...

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 2 ) ) E ( d ) ) = l g ( u ) E

...

Page 2233

100 9 THEOREM . The operator f ( T ) of Definition 8 is closed , linear , and

independent of the particular sequence of

each

f ( T ) ...

100 9 THEOREM . The operator f ( T ) of Definition 8 is closed , linear , and

independent of the particular sequence of

**Borel**sets used to define it . ( i ) Foreach

**Borel**set e and each x in Dif ( T ) ) , E ( e ) D ( f ( T ) ) S D ( f ( T ) ) and E ( e )f ( T ) ...

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