Sums of Bisectorial Operators and Applications

Abstract.

We study sums of bisectorial operators on a Banach space X and show that interpolation spaces between X and D(A) (resp. D(B)) are maximal regularity spaces for the problem Ay + By = x in X. This is applied to the study of regularity properties of the evolution equation u′ + Au = f on $\mathbb{R}$ for $f \in L^{p} {\left( {\mathbb{R};X} \right)}$ or $BUC{\left( {\mathbb{R};X} \right)},$ and the evolution equation u′ + Au = f on [0, 2π] with periodic boundary condition u(0) = u(2π) in $L^{p}_{{2\pi }} {\left( {\mathbb{R};X} \right)}$ or $C_{{2\pi }} {\left( {\mathbb{R};X} \right)}.$