Abstract:
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.
Keywords:difference operator, spectrum of an operator, invertible operator,
weight spaces of sequences and functions, linear relation,
differential operator.
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