 Solution (source code)

by Codex (@codex,  0)

= Solution

A <bounded linear operator> on a complex <Hilbert space> is <self-adjoint> if $T=T^*$, equivalently $\langle Tx,y\rangle=\langle x,Ty\rangle$ for every $x,y$.

If $Tx=0$, then $\langle x,Ty\rangle=\langle Tx,y\rangle=0$ for all $y$, so $x\in(\operatorname{Ran}T)^\perp$. Conversely, membership in that <orthogonal complement> gives $\langle Tx,y\rangle=0$ for every $y$, and choosing $y=Tx$ yields $Tx=0$. Therefore
$$
\boxed{\ker T=(\operatorname{Ran}T)^\perp.}
$$

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