Spectral multiplier theorems via $H^\infty$ calculus and $R$-bounds

Abstract: We prove spectral multiplier theorems for H\"ormander classes $\mathcal{H}^\alpha_p$ for 0-sectorial operators A on Banach spaces assuming a bounded $H^\infty(\Sigma_\sigma)$ calculus for some $\sigma \in (0,\pi)$ and norm and certain R-bounds on one of the following families of operators: the semigroup $e^{--zA}$ on $\mathbb{C}_+$, the wave operators $e^{isA}$ for $s \in \mathbb{R}$, the resolvent $(\lambda -- A)^{-1}$ on $\mathbb{C} \backslash \mathbb{R}$, the imaginary powers $A^{it}$ for $t \in \mathbb{R}$ or the Bochner-Riesz means $(1-A/u)^\alpha_+$ for $u > 0.$ In contrast to the existing literature we neither assume that A operates on an Lp scale nor that A is self-adjoint on a Hilbert space. Furthermore, we replace (generalized) Gaussian or Poisson bounds and maximal estimates by the weaker notion of R-bounds, which allow for a unified approach to spectral multiplier theorems in a more general setting. In this setting our results are close to being optimal. Moreover, we can give a characterization of the (R-bounded) $\mathcal{H}^\alpha_1$ calculus in terms of R-boundedness of Bochner-Riesz means.