A functional calculus for almost sectorial operators and applications to abstract evolution equations

Abstract.

In this paper, we construct a functional calculus for linear operators A whose spectrum lies in a sector of the complex plane and whose resolvent satisfies $\|(z-A)^{-1}\| \leq M\,|z|^{\gamma}$ for some $-1 $ < $\gamma $ < $ 0$ and all z outside the sector. By means of this functional calculus, we define complex powers and semigroups associated with A and some of its powers. Finally, these abstract results are applied to prove existence and uniqueness of classical solutions for three different types of abstract differential equations.