A discrete Perron--Ta Li type theorem for the dichotomy of evolution operators (Q882049)

Roughly speaking, an evolution family \(\{U(t,s)\}_{t\geq s\geq 0}\) of bounded operators in a Banach space \(X\) has the exponential dichotomy property if there is a family of projections \(\{P(t)\}_{t\geq 0}\) commuting with \(\{U(t,s)\}_{t\geq s\geq 0}\) such that \(U\) is exponentially stable on \(PX\) and \(U\) is invertible on the kernel of \(P\) with \(U^{-1}\) exponentially stable there. The main result of the present paper is a characterization of exponentially dichotomic evolution families in terms of their behaviour for \(t,s\) being natural numbers. More precisely, it is shown that a family is exponentially dichotomic if and only if the corresponding difference equation with inhomogeneity from \(l^p(X)\) admits a solution in \(l^q(X)\) with \(1/p+1/q\) not necessarily equal to 1. The advantage of this result is that it does not require any continuity hypotheses on \(\{U(t,s)\}_{t\geq s\geq 0}\).