  Let $A,$ $T$ and $B$ be bounded linear operators on a Banach space. This
paper is concerned mainly with finding some necessary and sufficient conditions
for convergence in operator norm of the sequences $\left\{ A^{n}TB^{n}\right\}
$ and $\left\{ \frac{1}{n}\sum_{i=0}^{n-1}A^{i}TB^{i}% \right\} $. These
results are applied to the Toeplitz, composition and model operators. Some
related problems are also discussed.
