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

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Page 1930

4 we have seen that a bounded normal operator in Hilbert space always has a

uniquely defined bounded and

defined on the field of all Borel subsets of the plane . The operators we shall

study in the ...

4 we have seen that a bounded normal operator in Hilbert space always has a

uniquely defined bounded and

**countably additive**resolution of the identitydefined on the field of all Borel subsets of the plane . The operators we shall

study in the ...

Page 2143

This spectral measure is bounded , is

commutes with T . PROOF . From Definitions 1 , 4 , and 7 it is seen that S ( T ) SS

( T ) , and thus for each & in S ( T ) there is , by Lemma 2 , one and only one

projection E ...

This spectral measure is bounded , is

**countably additive**on S ( T ) , andcommutes with T . PROOF . From Definitions 1 , 4 , and 7 it is seen that S ( T ) SS

( T ) , and thus for each & in S ( T ) there is , by Lemma 2 , one and only one

projection E ...

Page 2144

It clearly preserves finite disjoint unions , takes complements into complements ,

is

show that A ( 0 ) A ( S ) = A ( od ) . It is seen , by using the above remarks , that for

...

It clearly preserves finite disjoint unions , takes complements into complements ,

is

**countably additive**in the X topology of X * , and is bounded . It remains only toshow that A ( 0 ) A ( S ) = A ( od ) . It is seen , by using the above remarks , that for

...

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