Page 331 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes Thus M reduces T.
TP PTP and PT PTP ...(1)
Now suppose M reduces T. Then from (1), TP = PTP and PT = PTP. This gives TP = PT.

Conversely, let TP = PT
2
PTP =P T (Multiplying both sides on left by P.)
or PTP = PT  P 2 P

similarly multiplying both sides of TP = PT on the right of P, we get
2
TP = PTP or TP = PTP. Thus
TP = PT TP = PTP and PT = PTP.
Therefore from (1), we conclude that M reduces T.
Theorem 8: If M and N are closed linear subspace of a Hilbert space H and P and Q are the
projections on M and N respectively, then
(i) M N PQ O. and

(ii) PQ O QP O.
Proof: Since P and Q are projections on a Hilbert space H, therefore P* = P, Q* = Q.

We first observe that
PQ O PQ * O * Q * P* O *
QP O.

Therefore to prove the theorem it suffices to prove that
M N PQ O.

First suppose M N. If y is any vector in N, then M N y is orthogonal to every vector in M.

so y M .Consequently N M .

Now, let z be any vector in H. Then Qz is the range of Q i.e. Qz is in N.
Consequently Qz is in M which is null space of P.

Therefore P(Qz) = O.
Thus PQz = O z H

PQ = O
Conversely, let PQ = O and x M and y N.

since M is the range of P, therefore Px = x. Also N is the range of Q. Therefore
Qy = y

*
We have (x,y) = (Px, Qy) = (x,P Qy)

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