About the general solution of a linear homogeneous differential equation in a Banach space in the case of complex characteristic operators

A linear inhomogeneous differential equation (LIDE) of the $n$th order with constant bounded operator coefficients is studied in Banach space. Finding a general solution of LIDE is reduced to the construction of a general solution to the corresponding linear homogeneous differential equation (LHDE). Characteristic operator equation for LHDE is considered in the Banach algebra of complex operators. In the general case, when both real and complex operator roots are among the roots of the characteristic operator equation, the $n$-parametric family of solutions to LHDE is indicated. Operator functions $e^{A t},$ $\sin Bt,$ $\cos Bt$ of real argument $t \in {\,[0 ,\infty )}$ are used when building this family. The conditions under which this family of solutions form a general solution to LHDE are clarified. In the case when the characteristic operator equation has simple real operator roots and simple pure imaginary operator roots, a specific form of such conditions is indicated. In particular, these roots must commute with LHDE operator coefficients. In addition, they must commute with each other. In proving the corresponding assertion, the Cramer operator-vector rule for solving systems of linear vector equations in a Banach space is applied.