Green's function of the problem of bounded solutions in the case of a block triangular coefficient

It is known that the equation $x^{\mathrm{\prime }}(t)=Ax(t)+f(t)$, where $A$ is a bounded linear operator, has a unique bounded solution $x$ for any bounded continuous free term~$f$ if and only if the spectrum of the coefficient $A$ does not intersect the imaginary axis. The solution can be represented in the form \begin{equation*} x(t)=\int_{-\infty}^{\infty}\mathcal G(s)f(t-s),ds. \end{equation*} The kernel $\mathcal{G}$ is called Green’s function. In this paper, the case when $A$ admits a representation by a block triangular operator matrix is considered. It is shown that the blocks of $\mathcal{G}$ are sums of special convolutions of Green’s functions of diagonal blocks of $A$.