Normal operators with highly incompatible off-diagonal corners
Volume 256 / 2021
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 $D=\bigl [\begin {smallmatrix} D_1& D_2 \\ D_3 & D_4 \end {smallmatrix}\bigr ]$ relative to the decomposition $\mathcal H = \operatorname{ran} P \oplus \operatorname{ran} (I-P)$. 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 $\operatorname{rank} D_2 = j$ and $\operatorname{rank} D_3 = k$? Complete results are obtained in the case where $\dim \mathcal H \lt \infty $, and partial results are obtained in the infinite-dimensional setting.
