## Linear Operators: Spectral operators |

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Page 2169

This shows that ( vi ) holds for every

continuous function g . A repetition of this argument shows that it also holds if f

and g are both

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 f

and g are both

**bounded**Borel functions . Thus the**operators**f ( T ) and g ( T ' )commute ...

Page 2239

n + 00 n → n + 00 Since f xe is a bounded function , the operator T ( f Xe ) is a

operational calculus for bounded functions ( cf . XVII . 2 . 10 ) that T ( fxe ) = T ( fxe

) E ...

n + 00 n → n + 00 Since f xe is a bounded function , the operator T ( f Xe ) is a

**bounded operator**. If x is in Esē ) X as well as in E ( e ) X , it follows from theoperational calculus for bounded functions ( cf . XVII . 2 . 10 ) that T ( fxe ) = T ( fxe

) E ...

Page 2252

An attempt to follow the development in the

runs into difficulties . The

quasi - nilpotent restriction to each space E ( 0 ) X with o

An attempt to follow the development in the

**bounded**case by writing N = T - Sruns into difficulties . The

**operator**N , although easily seen by Lemma 2 to have aquasi - nilpotent restriction to each space E ( 0 ) X with o

**bounded**, need not be ...### What people are saying - Write a review

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