Page 317 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
T * T
1 1
T is self-adjoint.
1
1 *
Also T * T T *
2
2i
1
T T * *
2i

1
T * T * *
2i
1 1
T * T T T * T 2
2 2i

T * T
2 2
T is self-adjoint.
2
Thus T can be expressed in the form (1) where T ,T are self adjoint operators.
1 2
To show that (1) is unique.
Let T = U + iU , U ,U are both self-adjoint
1 2 1 2
We have T* U iU *
1 2
U * iU *
1 2
U * 1 iU * 2

U * 1 iU * 2 U 1 iU 2

T T* U 1 iU 2 U 1 iU 2 2U,

1
U T T * T
1 1
2
and T T* U iU U iU 2iU
1 2 1 2 2
1
U T T * T
2 2
2i
expression (1) for T is unique.
This completes the proof of the theorem.

Note The above result is analogous to the result on complex numbers that every complex
number z can be uniquely expressed in the form z = x + iy where x, y are real. In the above
theorem T =T + T , T is called real part of T and T is called the imaginary part of T.
1 2 1 2

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