GREEN'S FUNCTION OF THE PROBLEM OF BOUNDED SOLUTIONS IN THE CASE OF A BLOCK TRIANGULAR COEFFICIENT

It is well known that the equation x'(t) - Ax(t) F f(t), t is an element of R , where A is a bounded linear operator, has a unique bounded solution x for any bounded continuous free term f , provided the spectrum of the coefficient A does not intersect the imaginary axis. This solution can be represented in the form x(t) = integral(infinity)(-infinity) G(s)f(t - s) ds. The kernel 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 G are sums of special convolutions of Green's functions of the diagonal blocks of A.