In this paper we solve Sylvester matrix equation with infinitely-many solutions and conduct their classification. If the conditions for their existence are not met, we provide a way for their approximation by least-squares minimal-norm method. with solution X ∈ L(V 1 , V 2) are called Sylvester equations, Sylvester-Rosenblum equations or algebraic Ricatti equations. Such equations have various application in vast fields of mathematics, physics, computer science and engineering (see e.g. [5], [19] and references therein). Fundamental results, established by Sylvester and Rosenblum themselves, are now-days the starting point in solving contemporary problems where these equations occur. These results are Theorem 1.1. [23] (Sylvester matrix equation) Let A, B and C be matrices. Equation AX − XB = C has unique solution X iff σ(A) ∩ σ(B) = ∅. Theorem 1.2. [22] (Rosenblum operator equation) Let A, B and C be bounded linear operators. Equation AX − XB = C has unique solution X if σ(A) ∩ σ(B) = ∅. Equations with unique solutions have been extensively studied so far. There are numerous results regarding this case, some of them theoretical (e. g. Lyapunov stability criteria and spectral operators), which can be found in [2], [5] or [9], and some of them computational (matrix sign function, factorization of matrices and operators, various iterative methods etc.). It should be mentioned that matrix eq. (1) with unique solution X has been solved numerically (among others
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