On the Case of Complex Roots of the Characteristic Operator Polynomial of a Linear nth-Order Homogeneous Differential Equation in a Banach Space

We consider a linear homogeneous nth-order differential equation, n is an element of N, with constant bounded operator coefficients in a Banach space. Under some conditions on the (real and complex) roots of the corresponding characteristic equation, we obtain a formula expressing the general solution of the differential equation via the operator functions given by the exponential, sine, and cosine of the roots. The case in which n is even and the characteristic equation has n/2 pairs of complex-conjugate pure imaginary roots is investigated in detail. The case of a second-order differential equation is considered separately.