Paley-Littlewood decomposition for sectorial operators and interpolation spaces

We prove Paley-Littlewood decompositions for the scales of fractional powers of $0$-sectorial operators $A$ on a Banach space which correspond to Triebel-Lizorkin spaces and the scale of Besov spaces if $A$ is the classical Laplace operator on $L^p({\mathbb{R}}^n)\mathrm{.}$We use the $H^{\mathrm{\infty }}$-calculus, spectral multiplier theorems and generalized square functions on Banach spaces and apply our results to Laplace-type operators on manifolds and graphs, Schr"odinger operators and Hermite expansion.We also give variants of these results for bisectorial operators and for generators of groups with a bounded $H^{\mathrm{\infty }}$-calculus on strips.