Page 316 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 29: Normal and Unitary Operators

Notes
TT * T * T x,x 0 x

TT * x;x T * Tx,x x

T * x,T * x Tx,T * *x x

2 2
T * x Tx x  T * * T

T * x Tx x.

This completes the proof of the theorem.

2 2
Theorem 5: If N is normal operator on a Hilbert space H, then N N .
Proof: We know that if T is a normal operator on H then

Tx T * x x ...(1)

Replacing T by N, and x by Nx we get

NNx N * Nx x

2
N x N * Nx x ...(2)

2
Now N 2 Sup N x : x 1
Sup N * Nx : x 1 (by (2))

N * N
N 2

This completes the proof of the theorem.
Theorem 6: Any arbitrary operator T on a Hilbert space H can be uniquely expressed as
T T iT where T ,T 2 are self-adjoint operators on H.
1 2 1
T T * 1
Proof: Let T and T T T *
1 2
2 2i
Then T iT T ...(1)
1 2
1 *
Now T * T T *
1
2
1
T T * *
2
1
T * T * *
2
1 1
T * T T T * T 1
2 2

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