Let $X$ be a Banach space and $T$ be a bounded linear operator acting in $l_p(\mathbb Z^c,X)$, $1\le p\le\infty$. The operator $T$ is called \emph{ locally nuclear} if it can be represented in the form \begin{ equation*} (Tx)_k=\sum\limits_{ m\in\mathbb Z^c} b_{ km}x_{ k-m},\qquad k\in\mathbb Z^c, \end{ equation*} where $b_{ km}:\,X\to X$ are nuclear, \begin{ equation*} \lVert b_{ km}\rVert_{ \mathfrak S_1}\le\beta_{ m},\qquad k,m\in\mathbb Z^c, \end{ equation*} $\lVert\cdot\rVert_{ \mathfrak S_1}$ is the nuclear norm, $\beta\in l_{ 1}(\mathbb Z^c,\mathbb C)$ or $\beta\in l_{ 1,g}(\mathbb Z^c,\mathbb C)$, and $g$ is an appropriate weight on $\mathbb Z^c$. It is established that if $T$ is locally nuclear and the operator $\mathbf1+T$ is invertible, then the inverse operator $(\mathbf1+T)^{ -1}$ has the form $\mathbf1+T_1$, where $T_1$ is also locally nuclear. This result is refined for the case of operators acting in $L_p(\mathbb R^c,\mathbb C)$.
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