On the power-bounded operators of classes $C_{0 \\cdot}$ and $C_{1\n \\cdot}$

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