Exponential stability for abstract linear autonomous functional differential equations with infinite delay (Q1384240)

Summary: Based on the preceding paper [\textit{J. Liang} and \textit{T. Xiao}, Int. J. Math. Math. Sci. 14, No. 3, 497-508 (1991; Zbl 0743.34082)], this note is concerned with the exponential stability of the solution semigroup for the abstract linear autonomous functional-differential equation \[ \dot x(t)= L(x_t),\tag{\(*\)} \] where \(L\) is a continuous linear operator on some abstract phase space \(B\) into a Banach space \(E\). The authors prove that the solution semigroup of \((*)\) is exponentially stable if and only if the fundamental operator \((*)\) is exponentially stable and the phase space \(B\) is an exponentially fading memory space.