Page 328 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 30: Projections

M N R (PQ) Notes
Now let x R (PQ). Then (PQ)x = x
Now (PQ) x = x
P [(PQ) x] = Px

[P (PQ)] x = Px
(P Q) x = Px
2
(PQ) x = Px

But (PQ) x = x.
We have Px = x x M i.e. the range of P.
Also PQ = QP
x R (PQ) (PQ) x = x

(QP)x = x Q [(QP)x] = Qx
2
(Q P)x = Qx (QP)x = Qx
But (QP)x = x, Qx = x x N.

Thus x R (PQ) x M and x N
x M N
R(PQ) M N
Hence R(PQ) = M N.

Example: Show that an idempotent operator on a Hilbert space H is a projection on H
it is normal.
Solution: P is an idempotent operator on H i.e. P = P.
2
Let P be a projection on H. Then P* = P. We have
PP* = P* P* [taking P* in place of P in L.H.S.]
= P* P [ P* = P]
P is normal.

Conversely, let PP* = P*P.
Then to prove that P* = P.
For every vector y H, we have
(Py, Py) = (y, P* Py) = (y, PP*y) [ P*P = PP*]

= (P*y, P*y) [ (P*)* = P]
From this we conclude that
Py = 0 P*y = 0.

Now let x be any vector in H.
Let y = x – Px. Then

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