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.
Categories: math.FA math.OA