Page 318 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 29: Normal and Unitary Operators

Theorem 7: If T is an operator on a Hilbert space H, then T is normal its real and imaginary Notes
parts commute.

Proof: Let T and T be the real and imaginary parts of T. Then T , T are self-adjoint operators
1 2 1 2
and T = T + i T .
1 2
We have T* = (T + iT )* = T * + (iT )*
1 2 1 2
= T * + i T *
i 2
= T * – iT *
i 2
= T – iT
1 2
Now TT* = (T + iT ) (T – iT )
1 2 i 2
2
= T + T + i (T T – T T ) … (1)
2
1 2 2 1 1 2
and T*T = (T – iT ) (T – iT )
i 2 1 2
= T + T + i (T T – T T ) … (2)
2
2
1 2 1 2 2 1
Since T is normal i.e. TT* = T*T.
Then from (1) and (2), we see that
2
2
2
2
T + T + i (T T – T T ) = T + T + i (T T – T T )
1 2 2 1 1 2 1 2 1 2 2 1
T T – T T = T T – T T
2 1 1 2 1 2 2 1
2T T = 2T T
2 1 1 2
T T = T T T , T commute.
2 1 1 2 1 2
Conversely, let T , T commute
1 2
i.e. T T = T T , then from (1) and (2)
1 2 2 1
We see that
TT* = T*T T is normal.
Example: If T is a normal operator on a Hilbert space H and is any scalar, then T – I is
also normal.
Solution: T is normal TT* = T*T
Also (T – I)* = T* – ( I)*

= T* – I*

= T* – I.

Now (T – I) (T – I)* = (T – I) (T* – I)

= TT* – I – T* + | | I … (1)
2
Also (T – I)* (T – I) = (T* – I) (T – I)

= T*T – I* – T + | | I … (2)
2

LOVELY PROFESSIONAL UNIVERSITY 311