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

A local absolutely summing operator is an operator T acting in ${\mathrm{l}}_{\mathrm{p}}\left({\mathrm{\mathbb{Z}}}^{\mathrm{c}}\mathrm{,}\mathrm{X}\right)$, $1\le p\le \infty$, of the form

$(Tx{)}_k=\sum _{ m\in {\mathrm{\mathbb{Z}}}^c}{b}_{km}{x}_{k-m}\mathrm{,}k\in {\mathrm{\mathbb{Z}}}^c\mathrm{,}$

where X is a Banach space, b_km : X → X is an absolutely summation operator, and

$\parallel b_{km}{\parallel }_{\mathbf{AS}\left(X\right)}\le {\beta }_m$

for some $\beta \in l_1\left({\mathrm{\mathbb{Z}}}^{\mathrm{c}}\mathrm{,}\mathrm{ℂ}\right)$, $\parallel \cdot {\parallel }_{\mathbf{AS}\left(X\right)}$ is the the norm of the ideal of absolutely summing operators. We prove that if the operator 1+T is invertible, then the inverse operator has the form 1+T₁, where T₁ is also a local absolutely summing operator. A similar assertion is proved for the case where the operator T acts in $L_p\left({\mathrm{ℝ}}^{\mathrm{c}}\mathrm{,}\mathrm{ℂ}\right)$), $1\le p\le \infty$.