quantum mechanics - How to write "postselection" operator?

Suppose, I wish to know an operator, which eigenvalue is 1 if state is exactly F and 0 otherwise. I.e. this operator should select some desired state.

Which is this operator?

One of it's eigenfunctions is F, but what one is another?

Answer

All other functions not containing a non-zero factor of $F$ are eigenfunctions with eigenvalue $0$ of this operator, the eigenspace with that value is degenerate with dimension $\mathrm{dim}(\mathcal{H}) - 1$, where $\mathcal{H}$ is the whole space.

Obviously, the operator is simply the projection on $\lvert F \rangle$, so it is $P_F = \lvert F \rangle \langle F \rvert$.