L^sub p^-solvability of parabolic problems with an operator satisfying the Kato conjecture

We study the solvability in the spaces L ^sup p^(0, T;X) of an abstract parabolic equation with an operator defined by a sesquilinear form satisfying the Kato conjecture, where X is a Hilbert space obtained by interpolation between the domain of the form and the dual space. We describe the initial data spaces for various values of p and show that in the most common cases they coincide with the Besov spaces.