  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 $L^p(\mathbb{R}^n).$We use the $H^\infty$-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
$H^\infty$-calculus on strips.
