Let $A$ be a $0$-sectorial operator with a bounded $H^\infty(��\_��)$-calculus for some $��\in (0,��),$ e.g. a Laplace type operator on $L^p(��),\: 1 < p < \infty,$ where $��$ is a manifold or a graph. We show that $A$ has a H{��}rmander functional calculus if and only if certain operator families derived from the resolvent $(��- A)^{-1},$ 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 $L^p(��)$ space with $1 \leq p < \infty,$ $R$-boundedness reduces to well-known estimates of square sums.

Error in the title corrected