  We propose an approach based on perturbation theory to establish maximal
$L^p$-regularity for a class of integro-differential equations. As the left
shift semigroup is involved for such equations, we study maximal regularity on
Bergman spaces for autonomous and non-autonomous integro-differential
equations. Our method is based on the formulation of the integro-differential
equations to a Cauchy problems, infinite dimensional systems theory and some
recent results on the perturbation of maximal regularity (see \cite{AmBoDrHa}).
Applications to heat equations driven by the Dirichlet (or Neumann)-Laplacian
are considered.
