On the inverse closedness of the subalgebra of local absolutely summing operators

A local absolutely summing operator is an operator $T$ acting in $l_p(\mathbb{Z}^c,X)$, $1\le p\le\infty$, of the form
\begin{equation*} (Tx)_k=\sum_{m\in\mathbb{Z}^c}b_{km}x_{k-m}, \quad k\in\mathbb{Z}^c, \end{equation*}
where $X$ is a Banach space, $b_{km}\colon X\to X$ is an absolutely summation operator, and
\begin{equation*} \lVert b_{km}\rVert_{\mathbf A\mathbf S(X)}\le\beta_{m} \end{equation*}
for some $\beta\in l_{1}(\mathbb{Z}^c,\mathbb{C})$, $\lVert\cdot\rVert_{\mathbf{A}\mathbf{S}(X)}$ is the the norm of the ideal of absolutely summing operators. We prove that if the operator $\mathbf{1}+T$ is invertible, then the inverse operator has the form $\mathbf{1}+T_1$, where $T_1$ is also a local absolutely summing operator. A similar assertion is proved for the case where the operator $T$ acts in $L_p(\mathbb{R}^c,\mathbb{C})$, $1\le p\le\infty$.