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 -sectorial operators, which in turn is based on the notion of -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 -sectorial operator always has a bounded H (a)-functional calculus in its associated generalized Triebel-Lizorkin spaces.