Abstract
We suppose that M is a closed subspace of l
∞
(J, X), the space of all bounded sequences {x(n)}
n∈J
⊂ X, where J ∈ {Z
+
,Z} and X is a complex Banach space. We define the M-spectrum σ
M
(u) of a sequence u ∈ l
∞
(J,X). Certain conditions will be supposed on both M and σ
M
(u) to insure the existence of u ∈ M. We prove that if u is ergodic, such that σ
M
(u,) is at most countable and, for every λ ∈ σ
M
(u), the sequence e
−iλn
u(n) is ergodic, then u ∈ M. We apply this result to the operator difference equationu(n + 1) = Au(n) + ψ(n), n ∈ J,and to the infinite order difference equation Σ
r
k=1
a
k
(u(n + k) − u(n)) + Σ
s ∈ Z
ƒ(n − s)u(s) = h(n), n∈J, where ψ∈l
∞
(Z,X) such that ψ|
J
∈ M, A is the generator of a C
0
-semigroup of linear bounded operators {T(t)}
t>0
on X, h ∈ M, ƒ ∈ l
1
(Z) and a
k
∈C. Certain conditions will be imposed to guarantee the existence of solutions in the class M.