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}\leβ_{m},\qquad k,m\in\mathbb Z^c, \end{equation*} $\Vert \cdot {\Vert }_{{\mathfrak{S}}_1}$ 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 $1+T$ is invertible, then the inverse operator $(1+T{)}^{-1}$ has the form $1+T_1$, where $T_1$ is also locally nuclear. This result is refined for the case of operators acting in $L_p({\mathbb{R}}^c,\mathbb{C})$.

33 pages