  We study the Grushin operators acting on $\R^{d_1}_{x'}\times \R^{d_2}_{x"}$
and defined by the formula \[ L=-\sum_{\jone=1}^{d_1}\partial_{x'_\jone}^2 -
(\sum_{\jone=1}^{d_1}|x'_\jone|^2) \sum_{\jtwo=1}^{d_2}\partial_{x"_\jtwo}^2.
\] We obtain weighted Plancherel estimates for the considered operators. As a
consequence we prove $L^p$ spectral multiplier results and Bochner-Riesz
summability for the Grushin operators. These multiplier results are sharp if
$d_1 \ge d_2$. We discuss also an interesting phenomenon for weighted
Plancherel estimates for $d_1 <d_2$. The described spectral multiplier theorem
is the analogue of the result for the sublaplacian on the Heisenberg group
obtained by D. M\"uller and E.M. Stein and by W. Hebisch.
