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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

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    Probably orthogonal projection.2012-06-03
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    Do you need some other property, like self-adjoint?2012-06-03
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    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
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    I know what you say.2012-06-03
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    @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

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If "projection" is changed to "orthogonal projection", then here are some hints.

Hint 1: $\ker(T)^\perp=\{u\in H:Tx=0\Rightarrow \langle u,x\rangle=0\}$

Hint 2: Any $v\in H$ can be written as $v=u+k$ where $u\in\ker(T)^\perp$ and $k\in\ker(T)$.