On conditions for invertibility of difference and differential operators in weight spaces

We obtain necessary and sufficient conditions for the invertibility of the difference operator $\mathcal{D}_E\colon D(\mathcal{D}_E)\subset l^p_\alpha \to l^p_\alpha$, $(\mathcal{D}_E x)(n)=x(n+1)-Bx(n)$, $n\in \mathbb{Z}_+$, whose domain $D(\mathcal{D}_E)$ is given by the condition $x(0)\in E$, where $l^p_\alpha=l^p_\alpha(\mathbb{Z}_+,X)$, $p\in[1,\infty]$, is the Banach space of sequences (of vectors in a Banach space $X$) summable with weight $\alpha\colon\mathbb{Z}_+\to (0,\infty)$ for $p\in[1,\infty)$ and bounded with respect to $\alpha$ for $p=\infty$, $B\colon X\to X $ is a bounded linear operator, and $E$ is a closed $B$-invariant subspace of $X$. We give applications to the invertibility of differential operators with an unbounded operator coefficient (the generator of a strongly continuous operator semigroup) in weight spaces of functions.