Spectral multiplier theorems and averaged R-boundedness

Let $A$ be a $0$-sectorial operator with a bounded $H^\infty(\Sigma\_\sigma)$-calculus for some $\sigma \in (0,\pi),$ e.g. a Laplace type operator on $L^p(\Omega),\: 1 < p < \infty,$ where $\Omega$ is a manifold or a graph. We show that $A$ has a H{\"o}rmander functional calculus if and only if certain operator families derived from the resolvent $(\lambda - A)^{-1},$ the semigroup $e^{-zA},$ the wave operators $e^{itA}$ or the imaginary powers $A^{it}$ of $A$ are $R$-bounded in an $L^2$-averaged sense. If $X$ is an $L^p(\Omega)$ space with $1 \leq p < \infty,$ $R$-boundedness reduces to well-known estimates of square sums.