On the solution of the Yang–Baxter–like operator equation on Hilbert spaces
Hua Wang, Chang Liu, Junjie Huang · Linear and Multilinear Algebra · 2025
This paper deals with solutions of the Yang–Baxter–like operator equation AXA=XAX on the infinite dimensional Hilbert space H, where A is a rank-two bounded linear operator and X is the unknown bounded linear operator. For an operator X, necessary and sufficient conditions such that X is a solution of the equation AXA = XAX are given. With these criteria, it is in fact shown that all solutions of the nonlinear equation AXA = XAX under the corresponding conditions can be derived merely through solving some systems of linear equations.