## Linear Operators: Spectral operators |

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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 1931

+ 4 COROLLARY . If the domain of a

- field , then E is

The boundedness of E ( o ) follows from Corollaries IV . 10 . 2 and II . 3 . 21 .

+ 4 COROLLARY . If the domain of a

**countably additive**spectral measure E is a o- field , then E is

**countably additive**in the strong operator topology and bounded .The boundedness of E ( o ) follows from Corollaries IV . 10 . 2 and II . 3 . 21 .

Page 2144

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

is

show that A ( 0 ) A ( 8 ) = 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 ( 8 ) = A ( od ) . It is seen , by using the above remarks , that for

...

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