Hilbert space operators with compatible off-diagonal corners

Given a complex, separable Hilbert space H, we characterize those operators for which kP T(I − P)k = k(I − P)T Pk for all orthogonal projections P on H. When H is finite dimensional, we also obtain a complete characterization of those operators for which rank (I − P)T P = rank P T(I − P) for all orthogonal projections P. When 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.