  We study cosmological perturbations in the context of an interacting dark
energy model, where the holographic dark energy with IR cutoff decays into the
cold dark matter (CDM). For this purpose, we introduce three IR cutoffs of
Hubble horizon, particle horizon, and future event horizon. Here we present
small perturbations under the case that effective equation of state (EOS:
$\omega^{\rm eff}$) for the holographic energy density is determined to be the
same negative constant as that for the CDM. Such new matter productions tend to
dilute the density perturbations of CDM (matter contrast). For a decelerating
universe of $\omega^{\rm eff}>-1/3$, the matter contrast is growing as the
universe evolves, while for an accelerating universe of $\omega^{\rm
eff}<-1/3$, the matter contrast is decaying, irrespective of the choice of IR
cutoff. This shows clearly that the acceleration suppresses the growing of the
density perturbations at the early universe.
