Page 304 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 28: Self Adjoint Operators

Notes
2 A A  0 0, T * T and T T
n
0 as A A
n
A A * 0 A A* 0

A A* A is self adjoint

A S
S is closed.

Now since S is a closed linear subspace of the Banach space H , therefore S is a real Banach
space. ( S is a complete linear space)

Also I* I the identity operator I S.
This completes the proof of the theorem.

Theorem 3: If A ,A are self-adjoint operators, then their product A ,A is self adjoint
1 2 1 2
A ,A A ,A (i.e. they commute)
1 2 2 1
Proof: Let A ,A be two self adjoint operators in H.
1
2
Then A * A ,A * A .
1 1 2 2
Let A ,A commute, we claim that A ,A is self-adjoint.
2
2
1
1
A ,A * A * A * A A A A
1 2 2 1 2 1 1 2
A ,A * A A
1 2 1 2
A A 2 is self adjoint.
1
Conversely, let A A is self adjoint, then
1
2
A A * A A 2
2
1
1
A * A * A A 2
2
1
1
A ,A commute
1 2
This completes the proof of the theorem.
Theorem 4: If T is an operator on a Hilbert space H, then T = T 0 Tx,y 0 x,y H.
Proof: Let T = 0 (i.e. zero operator). Then for all x and y we have

T x,y Ox,y O,y O.

Conversely, Tx,y O x,y H

Tx,Tx O x,y H (taking y = Tx)

Tx O x,y H

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