Dichotomies for Band Matrices

Abstract

The bounded invertibility (as a linear map on $l_\infty $, say) of a bounded, strictly m-banded biinfinite matrix A is shown to be equivalent to a dichotomy or splitting of its kernel $\mathcal {N}$ (as a map on $\mathbb{R}^\mathbb{Z} $) into $\mathcal {N}^ + $ and $\mathcal {N}^ - $, with ,$\mathcal {N}^ + $ containing those which decay exponentially at $ + \infty $, and $\mathcal {N}^ - $ those which decay exponentially at $ - \infty $, 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 $A{\bf x} = {\bf b}$ by solutions of finite sections of this system is characterized in terms of linear independence, uniform as $I^ * \to - ( * \infty )$ of $\mathcal {N}$ over $I^ + \cup I^ - $, with $I^ * $ an integer interval of length $\mathcal {N}^ * , * = + , - $.