  Recent observations by Chandra and XMM-Newton demonstrate that the central
gas in "cooling flow" galaxy clusters has a mass cooling rate that decreases
rapidly with decreasing temperature. This contrasts the predictions of a steady
state cooling flow model. On the basis of these observational results, the gas
can be in a steady state only if a steady temperature dependent heating
mechanism is present; alternatively the gas could be in an unsteady state,
i.e., heated intermittently. Intermittent heating can be produced by accretion
on the supermassive black hole residing in the central cluster galaxy, via
Compton heating. This mechanism can be effective provided that the radiation
temperature of the emitted spectrum is higher than the gas temperature. Here we
explore whether this heating mechanism can be at the origin of the enigmatic
behavior of the hot gas in the central regions of ``cooling flow'' clusters.
Although several characteristics of Compton heating appear attractive in this
respect, we find that the fraction of absorbed heating for realistic gas and
radiation temperatures falls short by two orders of magnitude of the required
heating.
