Hilbert space operators with compatible off-diagonal corners

Abstract

Given a complex, separable Hilbert space $\mathcal{H}$, we characterize those operators for which $\Vert PT(I-P)\Vert =\Vert (I-P)TP\Vert$ for all orthogonal projections $P$ on $\mathcal{H}$. When $\mathcal{H}$ is finite-dimensional, we also obtain a complete characterization of those operators for which $\mathrm{rank}(I-P)TP=\mathrm{rank}PT(I-P)$ for all orthogonal projections $P$. When $\mathcal{H}$ is infinite-dimensional, we show that any operator with the latter property is normal, and its spectrum is contained in either a line or a circle in the complex plane.

24 pages