Suppose A is an arbitrary $n\times n$ matrix over the complex numbers. I am trying to find a nice characterization of the set of $n\times n$ matrices, $V = \{B:A^\dagger B+BA=0\}$.
I know that for the case of projections, $B=pp^\dagger$, this occurs precisely when $p$ is an eigenvector of $A$ with a pure imaginary eigenvalue. I was hopeful that these projections would span $V$ (which is a vector space over $\mathbb{C}$), but I don't know whether that is true. Any ideas?