## Linear Operators: Spectral operators |

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Page 1977

The

by equation ( ii ) is a countably additive spectral measure in HP defined on the

Borel sets B in the complex plane . Since E ( 0 ; Â ( s ) ) and Â ( s ) commute for ...

The

**preceding**theorem shows that the operator valued measure E ( o ; A ) givenby equation ( ii ) is a countably additive spectral measure in HP defined on the

Borel sets B in the complex plane . Since E ( 0 ; Â ( s ) ) and Â ( s ) commute for ...

Page 2232

Let { en } be as in the

E ( en ) xc = E ( e ) x , and lim QE ( en ) E ( e ) x = lim QE ( ene ) x n + 00 n + 00 =

lim E ( ene ) Qx = E ( e ) Qx n 00 by the second paragraph of the

Let { en } be as in the

**preceding**proof , and let x be in D ( Q ) . Then limn + . E ( e )E ( en ) xc = E ( e ) x , and lim QE ( en ) E ( e ) x = lim QE ( ene ) x n + 00 n + 00 =

lim E ( ene ) Qx = E ( e ) Qx n 00 by the second paragraph of the

**preceding**...Page 2396

Let 04 be as in the

) = A ( 04 ( : , ula ) ) ) ( cf . Lemma 4 for the definition of ula ) ) . Then , by the

as ...

Let 04 be as in the

**preceding**lemma , put A ( a ) = A ( 01 ( : , u ( ) ) ) , and let B ( a) = A ( 04 ( : , ula ) ) ) ( cf . Lemma 4 for the definition of ula ) ) . Then , by the

**preceding**lemma , by Lemma 1 , and by formulas ( 2a ) and ( 2b ) , ( B ( ) ~ | A ) |as ...

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