On the inverse of the sum of two sectorial operators

We study an abstract linear operator equation on a Banach space by using the inverse of the sum of two sectorial operators. We prove that the boundedness of a special type of operator valued H∞-calculus is sufficient for maximal regularity of the solution. We apply the result to the abstract parabolic problem, to give a maximal Lp-regularity condition. We also study the abstract hyperbolic problem and give a sufficient condition for the existence of solution.