Normal solvability of linear differential equations in the complex plane

The operator $L_nY=A(z)Y'(z)+B(z)Y(z)$, where $A(z)$ and $B(z)$ are square $n$th order matrices, regular in a region $G$ of arbitrary connectivity, and $Y(z)$ is a single-column matrix, regular in $G$, is investigated. The operator $L_nY$ is shown to be normally solvable in the space $A^n(G)$ of single-column matrices regular in $G$, and in certain subspaces of $A^n(G)$, and its index is evaluated.