Inverse-closedness of the subalgebra of locally nuclear operators

Let X be a Banach space and T be a bounded linear operator acting in l_p(ℤ^c,X), 1 ≤ p ≤ ∞. The operator T is called locally nuclear if it can be represented in the form

$${(Tx)_k} = \sum\limits_{m \in {\mathbb{Z}^c}} {{b_{km}}} {x_{k - m}},\quad k \in {\mathbb{Z}^c},$$

where b_km: X → X are nuclear,

$${\left\| {{b_{km}}} \right\|_{{\mathfrak{S}_1}}} \le {\beta _m},\quad k,m \in {\mathbb{Z}^c},$$

\(\left\|\cdot\right\|{_{{\mathfrak{S}_1}}}\) is the nuclear norm, β ∈ l₁(ℤ^c,ℂ) or β ∈ l_1,g(ℤ^c,ℂ), and g is an appropriate weight on 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 + T₁, where T₁ is also locally nuclear. This result is refined for the case of operators acting in L_p (ℝ^c,ℂ).