A General Proof of the Fong-Tsui Conjecture

Abstract

We present a general proof of the Fong-Tsui conjecture for bounded operators on arbitrary complex Hilbert spaces. Specifically, we show that $|T|\le |\mathrm{Re}T|$ implies that $T$ is self-adjoint. The argument combines a positive inverse of a Sylvester map with a spectral cutoff determined by the norm of the positive defect $|\mathrm{Re}T|-|T|$. A local vanishing lemma reduces the analysis to the classical squared self-adjointness criterion, while positivity of the defect yields a global norm contradiction. We formulate the argument as an abstract four-operator vanishing principle, without compactness, trace, or separability assumptions. We also establish a quantitative stability estimate: if $|T|\le |\mathrm{Re}T|+\epsilon I$ and $0\le \epsilon \le |T|$, then $|\mathrm{Im}T|\le 6|T{|}^{7/8}{\epsilon }^{1/8}$. The constant is independent of the dimension, and the exponent is not claimed to be optimal. Large language models (LLMs) were used to assist with proof development, algebraic calculations, numerical checks, and auditing of the arguments.

This preprint has been superseded by the joint paper arXiv:2609.16236 ("A proof of the Fong--Tsui conjecture", Aouichaoui--Kittaneh--Ma), which presents the same result. This earlier version is withdrawn to avoid duplicate posting; readers are referred to arXiv:2609.16236