Hilbert space operators with compatible off-diagonal corners

TL;DR

This paper characterizes operators on Hilbert spaces with specific symmetry properties related to off-diagonal corners, revealing conditions under which these operators are normal and describing their spectral properties.

Contribution

It provides a new characterization of Hilbert space operators with compatible off-diagonal corners, including a complete finite-dimensional classification and spectral analysis in infinite dimensions.

Findings

Operators with equal off-diagonal corner norms are characterized.

Finite-dimensional operators with rank conditions are fully classified.

Infinite-dimensional operators with rank conditions are normal with spectra on a line or circle.

Abstract

Given a complex, separable Hilbert space $\mathcal{H}$, we characterize those operators for which $\| P T (I-P) \| = \| (I-P) T P \|$ 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 $\mathrm{rank}\, (I-P) T P = \mathrm{rank}\, P T (I-P)$ 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.