Yang-Baxter算子方程的解及其表示

doi:10.12677/aam.2026.158331

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Yang-Baxter算子方程的解及其表示
Solutions and Representations of the Yang-Baxter Operator Equation

DOI: 10.12677/aam.2026.158331, PDF, 科研立项经费支持
作者: 张丽霞, 王华^*：内蒙古工业大学理学院，内蒙古呼和浩特
关键词: Yang-Baxter算子方程；算子方程组；算子矩阵；幂等算子；Yang-Baxter Operator Equation； System of Operator Equations； Operator Matrix； Idempotent Operator

摘要: 本文研究Hilbert空间中Yang-Baxter算子方程

A X A = X A X

解的存在性以及解的表示。首先，建立Yang-Baxter算子方程与线性算子方程组之间的联系，将非线性的Yang-Baxter算子方程的解转化为线性算子方程组的解的问题。其次，利用算子的矩阵表示与广义逆理论，刻画该线性算子方程组的可解性，进而给出解的显示表达式。以此为基础，给出Yang-Baxter算子方程的解存在的条件以及解的广义逆表示和矩阵形式的表示。

Abstract: This paper investigates the existence and explicit representation of solutions to the Yang-Baxter operator equation in Hilbert spaces. First, we establish a connection between the nonlinear Yang-Baxter operator equation and a system of linear operator equations, thereby transforming the solution problem of the former into that of the latter. Next, using matrix representations of operators and generalized inverse theory, we characterize the solvability of the linear system and derive explicit expressions for its solutions. Based on these results, necessary and sufficient conditions for the solvability of the Yang-Baxter operator equation are obtained, and representations of its solutions are given in terms of both generalized inverses and matrix forms.

文章引用：张丽霞, 王华. Yang-Baxter算子方程的解及其表示[J]. 应用数学进展, 2026, 15(8): 44-51. https://doi.org/10.12677/aam.2026.158331

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