Page 332 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 30: Projections

Notes
= (x,PQy)  P* P
= (x,Oy) PQ O

= (x,O) = O
M and N are orthogonal i.e. M N.

30.1.3 Orthogonal Projections

Definition: Two projections P and Q on a Hilbert space H are said to be orthogonal if PQ = O.
Note: By theorem 8, P and Q are orthogonal iff their ranges M and N are orthogonal.

Theorem 9: If P , P , ... P are projections on closed linear subspaces M , M , ... M of a Hilbert
1 2 n 1 2 n
space H, then P = P + P + ... + P is a projection the P's are pair-wise orthogonal (in the sense
1 2 n i
that PP 0,i j).
i j
Also then P is the projection on M = M + M + ... + M .
1 2 n
Proof: Given that P , P , ... P are projections on H.
1 2 n
*
Therefore P 2 P P for each i = 1, 2, ...,n.
i i i
Let P = P + P + ... + P . Then P = (P + P + ... + P ) = P * ... P *
*
*
1 2 n 1 2 n 1 n
= P + P + ... + P = P.
1 2 n
Sufficient Condition:
Let PP O,i j. Then to prove that
i j
P is a projection on H. We have

P = PP = (P + P + ... + P ) (P + P + ... + P )
2
1 2 n 1 2 n
2
= P 2 P ... P 2 PP 0,i j
1 2 n i j
= P + P + ... + P
1 2 n
= P
Thus, P* = P = P*.
Therefore P is a projection on H.
Necessary Condition:

Let P is a projection on H.
2
Then P = P = P*.
We are to prove that PP 0 if i j.
i j
We first observe that if T is any projection on H and z is any vector in H, then
(Tz, z) = (T Tz, z) = (Tz, T*z)
= (Tz, Tz)

2
= Tz ...(1)

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