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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 onL  p    (   R  n   )   .    We use the    H  ∞   ‐calculus, spectral multiplier theorems and generalized square functions on Banach spaces and apply our results to Laplace‐type operators on manifolds and graphs, Schrödinger operators and Hermite expansion. We also give variants of these results for bisectorial operators and for generators of groups with a bounded    H  ∞   ‐calculus on strips.

Paley–Littlewood decomposition for sectorial operators and interpolation spaces