Inverse-closedness of the subalgebra of locally nuclear operators

Abstract

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 $L_p(\mathbb R^c,\mathbb C)$.