## Linear Operators: Spectral theory |

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Page 925

16 Let N ,, N ,, ... be a countable

commuting with each other . Show that there exists a single Hermitian operator T

such that each Nx is a Borel function of T. ( Hint : Use Theorem 2.1 and Exercise

15 ) .

16 Let N ,, N ,, ... be a countable

**sequence**of normal operators in H , allcommuting with each other . Show that there exists a single Hermitian operator T

such that each Nx is a Borel function of T. ( Hint : Use Theorem 2.1 and Exercise

15 ) .

Page 959

prove the uniqueness of the limit it will suffice to show that if Molebn ) 2 k for

some n , then , for every e > 0 , Moleem ) > kếE for some m . Since Ueem = e , the

ebn .

prove the uniqueness of the limit it will suffice to show that if Molebn ) 2 k for

some n , then , for every e > 0 , Moleem ) > kếE for some m . Since Ueem = e , the

**sequence**{ eembno m 2 1 } is an increasing**sequence**of sets whose union isebn .

Page 1124

If E ,, E are in F and q ( En ) increases to the limit 9 ( E ) , then it follows from what

we have already proved that E is an increasing

SE . If Ec is the strong limit of En , then E. SE and Q ( E ) = Q ( E ) . Thus , it follows

...

If E ,, E are in F and q ( En ) increases to the limit 9 ( E ) , then it follows from what

we have already proved that E is an increasing

**sequence**of projections and E ,SE . If Ec is the strong limit of En , then E. SE and Q ( E ) = Q ( E ) . Thus , it follows

...

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