Rs-sectorial operators and generalized Triebel-Lizorkin spaces

We introduce a notion of generalized Triebel-Lizorkin spaces associated with sectorial operators in Banach function spaces. Our approach is based on holomorphic functional calculus techniques. Using the concept of Rₛ-sectorial operators, which in turn is based on the notion of Rₛ-bounded sets of operators introduced by Lutz Weis, we obtain a neat theory including equivalence of various norms and a precise description of real and complex interpolation spaces. Another main result of this article is that an Rₛ-sectorial operator always has a bounded H^∞-functional calculus in its associated generalized Triebel-Lizorkin spaces.