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.

Question

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.

ustalovatfog · Accepted Answer

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.

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.

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