## Linear Operators: Spectral operators |

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Page 1979

It follows from equations ( iv ) and ( v ) of Lemma 3 that E y ( s ) ; Â ( s ) ) is e -

essentially bounded on S . Lemma 4 then

is satisfied . Q . E . D . 8 COROLLARY . Every operator A in AP is the strong limit

of ...

It follows from equations ( iv ) and ( v ) of Lemma 3 that E y ( s ) ; Â ( s ) ) is e -

essentially bounded on S . Lemma 4 then

**shows**that condition ( i ) of the theoremis satisfied . Q . E . D . 8 COROLLARY . Every operator A in AP is the strong limit

of ...

Page 2169

This

continuous function g . A repetition of this argument

and g are both bounded Borel functions . Thus the operators f ( T ) and g ( T ' )

commute ...

This

**shows**that ( vi ) holds for every bounded Borel function f and everycontinuous function g . A repetition of this argument

**shows**that it also holds if fand g are both bounded Borel functions . Thus the operators f ( T ) and g ( T ' )

commute ...

Page 2170

These lemmas will

expansion of the scalar product ( ( aI – T ' ) x , ( QI — T ' ) x )

x12 = ] ( Q ) x12 + | ( R ( Q ) I — T ' ) x12 2 \ I ( Q ) | 2 | 2 / 2 , so that llaI – T ) 121 ...

These lemmas will

**show**that the hypotheses of Theorem 5 . ... If & is not real , anexpansion of the scalar product ( ( aI – T ' ) x , ( QI — T ' ) x )

**shows**that l ( a — T ' )x12 = ] ( Q ) x12 + | ( R ( Q ) I — T ' ) x12 2 \ I ( Q ) | 2 | 2 / 2 , so that llaI – T ) 121 ...

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