Page 334 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 30: Projections

2 2 Notes
Px x Px x
i i i i
Px x
i i
x the range of P.
i
x R P , for each i 1,2,...,n
i
x x ... x R P .
1 2 n
x R P .
Then x M x R P

M R P ...(3)

Now suppose that x R P . Then
Px = x

P P ... P x x
1 2 n
P x P x ... P x x
1 2 n
But P x M ,P x M ,...,P x M .
1 1 2 2 n n
x M M ... M and so R P M ...(4)
1 2 n
Hence from (3) and (4), we get
M = R(P)

This completes the proof of the theorem.

30.2 Summary

 A projection P on a Hilbert space H is said to be a perpendicular projection on H if the
range M and null space N of P are orthogonal.

 Let T be an operator on a Hilbert space H and M be a closed subspace of H. Then M is said
to be invariant under T if T M M.

 Let T be an operator on Hilbert space H, if M is closed subspace of H and if M and M are
both invariant under T, then T is said to be reduced by M.
 Two projections P and Q on a Hilbert space H are said to be orthogonal if PQ = O.

30.3 Keywords

Invariance: Let T be an operator on a Hilbert space H and M be a closed subspace of H. Then M
is said to be invariant under T if T M M.

Orthogonal Projections: Two projections P and Q on a Hilbert space H are said to be orthogonal
if PQ = O.
Perpendicular Projections: A projection P on a Hilbert space H is said to be a perpendicular
projection on H if the range M and null space N of P are orthogonal.

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