1
$\begingroup$

My question is:

Let $H$ be a Hilbert space and $T \in B(H)$. Prove that $T$ is a projection if and only if $T$ is the identity on the orthogonal complement of its kernel.

Thanks

  • 0
    Probably orthogonal projection.2012-06-03
  • 0
    Do you need some other property, like self-adjoint?2012-06-03
  • 1
    I assume "$T$ is a projection" is defined to mean "$T = T^*$ and $T^2 = T$." In this case, here is a hint (mainly for the "if" direction, which is the harder one): for any $A$ in $\mathcal{B}(H)$, the orthogonal complement of the kernel of $A$ is the closure of the range of $A^*$.2012-06-03
  • 0
    I know what you say.2012-06-03
  • 1
    @leslietownes: If $T^*=T$ and $T^2=T$, then $T$ is an orthogonal projection (which is indeed what the question should specify).2012-06-03

1 Answers 1