Exponentially dichotomous generators of evolution bisemigroups on admissible function spaces (Q2901923)

To an evolution family \(\mathcal{U} = (U(t,s))_{t\geq s\geq 0}\) of bounded operators on a Banach space \(X\) and through the integral equation NEWLINE\[NEWLINEu(t) = U(t,s)u(s) +\int_s^t U(t, \xi)f(\xi)d\xi,NEWLINE\]NEWLINE the authors associate an operator \(G_Z\) acting on Banach space of \(X\)-valued functions corresponding to admissible Banach function spaces. These function spaces contain \(L_p\) spaces (\(1\leq p < \infty\)), Lorentz spaces \(L_{p,q}\) and many other function spaces obtained from interpolation theory. The authors show that the exponential dichotomy of \(\mathcal{U}\) is equivalent to the exponential dichotomy of the operator \(G_Z\) generating a bisemigroup \((\mathcal{T}(t))_{t\in \mathbb{R}}\). They also prove that the exponential dichotomy of \(G_Z\) is robust under small perturbations by bounded operators. This leads to an application to vector-valued Wiener-Hopf and Riccati equations.