Some Convergence Theorems for Operator Sequences

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 {A(n)TB(n)} and {1/n Sigma(n-1)(i=0) A(i)TB(i)}. These results are applied to the Toeplitz, composition, and model operators. Some related problems are also discussed.