  We characterize the polynomial decay of orbits of Hilbert space
$C_0$-semigroups in resolvent terms. We also show that results of the same type
for general Banach space semigroups and functions obtained recently in the
paper by C.J.K.Batty and T.Duyckaerts, Non-uniform stability for bounded
semi-groups on Banach spaces (J. Evol. Eq. 2008), are sharp. This settles a
conjecture posed in the paper by C.J.K.Batty and T.Duyckaerts.
