Dichotomies for Band Matrices

The bounded invertibility (as a linear map on l_∞, say) of a bounded, strictly m-banded biinfinite matrix A is shown to be equivalent to a dichotomy or splitting of its kernel N (as a map on RZ) into N^ + and N^ -, with ,N^ + containing those which decay exponentially at + ∞, and N^ - those which decay exponentially at - ∞, together with a certain uniformity (with respect to the sequence index) of this direct sum decomposition. The approximability of the solution of the biinfinite system A x = b by solutions of finite sections of this system is characterized in terms of linear independence, uniform as I^ * → - ( * ∞ ) of N over I^ + ∪ I^ -, with I^ * an integer interval of length N^ * , * = + , -.