This paper is devoted to the study of the difference equation \((1)\quad x_{k+1}=A_ kx_ k\) \((k=0,1,2,...)\) over some real or complex Banach space B. Here \(A_ k\) is a bounded linear operator from B to B. Suppose that for every non-negative integer k the set \(M_ k\) of all elements x of B for which \(\{\) \(| | x| |,| | A_ kx| |,| | A_{k+1}A_ kx| |,...\}\) is bounded is a closed complemented subspace of B and that \(L_ k\) is a subspace of B such that B is the direct sum of \(M_ k\) and \(L_ k.\)
(1) is said to possess an exponential dichotomy if there exist constants \(a>0\) and \(b\in (0,1)\) such that for all nonnegative integers k,l,m with \(k\geq l\geq m\) and for all \(x\in B\) both \(| | A_{k-1}A_{k- 2}\cdot...\cdot A_ mP_{M_ m}x| | \leq ab^{k-1}| | A_{l-1}A_{l-2}\cdot...\cdot A_ mP_{M_ m}x| |\) and \(| | A_{l-1}A_{l-2}\cdot...\cdot A_ mP_{L_ m}x| | \leq ab^{1-k}| | A_{k-1}A_{k-2}\cdot...\cdot A_ mP_{L_ m}x| |\) hold. Here \(P_{M_ m}\) and \(P_{L_ m}\) denote the projections onto \(M_ m\) and \(L_ m\), respectively.
Necessary (respectively sufficient) conditions are provided for the fact that (1) possesses an exponential dichotomy.