Geodesic
On conditions for invertibility of difference and differential operators in weight spaces
Izvestiya. Mathematics, Tome 75 (2011) no. 4, pp. 665-680

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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. 
M. S. Bichegkuev. On conditions for invertibility of difference and differential operators in weight spaces. Izvestiya. Mathematics, Tome 75 (2011) no. 4, pp. 665-680. http://geodesic.mathdoc.fr/item/IM2_2011_75_4_a0/
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