  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.
