Normal operators with highly incompatible off-diagonal corners

Abstract: Let $\mathcal{H}$ be a complex, separable Hilbert space, and $\mathcal{B}(\mathcal{H})$ denote the set of all bounded linear operators on $\mathcal{H}$. Given an orthogonal projection $P \in \mathcal{B}(\mathcal{H})$ and an operator $D \in \mathcal{B}(\mathcal{H})$, we may write $D=\begin{bmatrix} D_1& D_2 D_3 & D_4 \end{bmatrix}$ relative to the decomposition $\mathcal{H} = \mathrm{ran}\, P \oplus \mathrm{ran}\, (I-P)$. In this paper we study the question: for which non-negative integers $j, k$ can we find a normal operator $D$ and an orthogonal projection $P$ such that $\mathrm{rank}\, D_2 = j$ and $\mathrm{rank}\, D_3 = k$? Complete results are obtained in the case where $\mathrm{dim}\, \mathcal{H} < \infty$, and partial results are obtained in the infinite-dimensional setting.