  By a bounded backward sequence of the operator $T$ we mean a bounded sequence
$\{x_n\}$ satisfying $Tx_{n+1}=x_n$. In \cite{Pa} we have characterized
contractions with strongly stable nonunitary part in terms of bounded backward
sequences. The main purpose of this work is to extend that result to
power-bounded operators. Aditionally, we show that a power-bounded operator is
strongly stable ($C_{0 \cdot} $) if and only if its adjoint does not have any
nonzero bounded backward sequence. Similarly, a power-bounded operator is
non-vanishing ($C_{1 \cdot} $) if and only if its adjoint has a lot of bounded
backward sequences.
