New Results on the Solvability of Sylvester-Type Operator Equations

Abstract

This paper investigates several forms of Sylvester-type operator equations in infinite-dimensional Hilbert spaces, focusing on both the classical equation $AX-XB=C$ and its generalized version $AX-YB=C$, which involves two unknowns. We establish new necessary and sufficient conditions for the existence of solutions by employing generalized inverses under novel structural assumptions. Special attention is given to the behavior of these equations when restricted to subspaces such as $\ker(A+I)$ and $\ker(B+I)$, and to cases involving two distinct subspaces. The study highlights how operator properties-such as involution and pseudo-inverses-govern solvability and solution structure. The results offer a unified theoretical framework that encompasses both classical and generalized operator equations, with potential applications in control theory, perturbation analysis, and related areas. Illustrative examples are provided to demonstrate the applicability and relevance of the theoretical developments.