Let H be an Hilbert space and let M,N closed subspaces of H. LetP_M and P_N orthogonal projections with range(M) and range(N) respectively. Prove that P_M+P_N=P_M+N if and only if M_|_N.

Dylan Benitez

Dylan Benitez

Answered question

2022-11-03

Let H be an Hilbert space and let M , N closed subspaces of H. Let P M and P N orthogonal projections with range( M) and range( N) respectively. Prove that P M + P N = P M + N if and only if M N.

Answer & Explanation

ustalovatfog

ustalovatfog

Beginner2022-11-04Added 11 answers

Suppose first that M N. Then, for any x , y H,
P M P N x , y = P N x , P M y = 0.
As we can do this for all x , y we get that P M P N = 0 . Then
( P M + P N ) 2 = P M 2 + P N 2 = P M + P N
and P N + P M is a projection. Given x + y M + N with x M and y N,
( P M + P N ) ( x + y ) = P M x + P N x + P M y + P N y = x + y ,
since P M x = x, P n x = 0, P M y = 0, P N y = y. If z ( M + N ) , then
( P M + P N ) z , w = z , ( P M + P N ) w = 0.
So P M + P N = P M + N .
Conversely, if P M + P N = P M + N , then
P M + P N = P M + N = P M + N 2 = P M + P N + P N P M + P M P N .
Then
(*) P M P N + P N P M = 0.
Multiply on the left by I P M , and we get ( I P M ) P N P M = 0. This is P M P N P M = P N P M . Then
P M P N = ( P N P M ) = ( P M P N P M ) = P M P N P M = P N P M .
Then P N P M = P M P N and now ( ) becomes P M P N = 0. If we take x M and y N,
x , y = P M x , P N y = P N P M x , y = 0.
So M N.

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