  The equation of state characterizing the dark energy component is constrained
by combining Chandra observations of the X-ray luminosity of galaxy clusters
with independent measurements of the baryonic matter density and the latest
measurements of the Hubble parameter as given by the HST key project. By
assuming a spatially flat scenario driven by a "quintessence" component with an
equation of state $p_x = \omega \rho_x$ we place the following limits on the
cosmological parameters $\omega$ and $\Omega_{\rm{m}}$: (i) $-1 \leq \omega
\leq -0.55$ and $\Omega_{\rm m} = 0.32^{+0.027}_{-0.014}$ (1$\sigma$) if the
equation of state of the dark energy is restricted to the interval $-1 \leq
\omega < 0$ (\emph{usual} quintessence) and (ii) $\omega =
-1.29^{+0.686}_{-0.792}$ and $\Omega_{\rm{m}} = 0.31^{+0.037}_{-0.034}$
($1\sigma$) if $\omega$ violates the null energy condition and assume values $<
-1$ (\emph{extended} quintessence or ``phantom'' energy). These results are in
good agreement with independent studies based on supernovae observations,
large-scale structure and the anisotropies of the cosmic background radiation.
