  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
$\mathcal{R}_s$-sectorial operators, which in turn is based on the notion of
$\mathcal{R}_s$-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 $\mathcal{R}_s$-sectorial operator always has a bounded
$H^\infty$-functional calculus in its associated generalized Triebel-Lizorkin
spaces.
