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 $D=\left[\begin{array}{ccc}{D}_1& D_2{D}_3& D_4\end{array}\right]$ 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 $\dim \mathcal{H}<\infty$, and partial results are obtained in the infinite-dimensional setting.

submitted