Page 315 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
Also by hypothesis, we have N N* =N* N and N N* N* N … (2)
1 2 2 1 2 1 1 2
we claim that N + N is normal.
1 2

i.e. (N + N )(N + N )* = (N + N )*(N + N ) ...(3)
1 2 1 2 1 2 1 2
since adjoint operation preserves addition, we have

(N + N )(N + N )* = (N + N ) N* +N*
1 2 1 2 1 2 1 2
N N* N N N N* N N* ...(4)
1 1 1 2 2 1 2 2
N N* 1 N* N 1 N* N 2 N* N 2
1
2
1
2
= N* N* N N
1 2 1 2
= N N * N N (using (1) and (2))
1 2 1 2
N 1 N 2 N 1 N * N 1 N * N 1 N 2
2
2
N N is normal.
1 2
Now we show that N N is normal i.e.
1 2
N N N N * N N * N N .
1 2 1 2 1 2 1 2

L.H.S.= N N 2 N N * N N N * N * 1
1
2
2
2
1
1
N N N * N *
1 2 2 1
N N * N N *
1 2 2 1
N N * 2 N N * 1
1
2
N * N N * N
2 1 1 2
N * N N * N
2 1 1 2
N * N * 1 N N 2
1
2
N N * N N
1 2 1 2
N N N N * N N * N N
1 2 1 2 1 2 1 2
N N is normal.
1 2
This completes the proof of the theorem.

Theorem 4: An operator T on a Hilbert space H is normal T * x Tx for every x H.

Proof: We have T is normal TT* T * T
TT * T * T 0

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