AI 中文总结
本文通过结合Sylvester映射正逆、谱截断和局部消失引理，给出Fong-Tsui猜想的一般性证明，并建立定量稳定性估计，无需紧性、迹或可分性假设。
AI 中文摘要
我们给出了任意复Hilbert空间上有界算子的Fong-Tsui猜想的一般性证明。具体地，我们证明$|T|\leq|\operatorname{Re}T|$蕴含$T$是自伴的。该论证将Sylvester映射的正逆与由正缺陷$|\operatorname{Re}T|-|T|$的范数确定的谱截断相结合。一个局部消失引理将分析归结为经典的平方自伴性判据，而缺陷的正性产生全局范数矛盾。我们将该论证表述为一个抽象的四算子消失原理，无需紧性、迹或可分性假设。我们还建立了一个定量稳定性估计：若$|T|\leq|\operatorname{Re}T|+\varepsilon I$且$0\leq\varepsilon\leq|T|$，则$|\operatorname{Im}T|\leq6|T|^{7/8}\varepsilon^{1/8}$。该常数与维数无关，且指数并非声称最优。大型语言模型（LLMs）被用于辅助证明开发、代数计算、数值检验和论证审计。
英文摘要
We present a general proof of the Fong-Tsui conjecture for bounded operators on arbitrary complex Hilbert spaces. Specifically, we show that $|T|\leq|\operatorname{Re}T|$ 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 $|\operatorname{Re}T|-|T|$. 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 $|T|\leq|\operatorname{Re}T|+\varepsilon I$ and $0\leq\varepsilon\leq|T|$, then $|\operatorname{Im}T|\leq6|T|^{7/8}\varepsilon^{1/8}$. 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.
CommentsThis preprint has been superseded by the joint paper arXiv:2609.16236 ("A proof of the Fong--Tsui conjecture", Aouichaoui--Kittaneh--Ma), which presents the same result. This earlier version is withdrawn to avoid duplicate posting; readers are referred to arXiv:2609.16236
