Inverse-closedness of the subalgebra of locally nuclear operators

Let X be a Banach space and T bounded linear operator acting in lp(ℤc,X), 1 ≤ p ∞. The is called locally nuclear if it can represented the form $${(Tx)_k} = \sum\limits_{m \in {\mathbb{Z}^c}} {{b_{km}}} {x_{k - m}},\quad k {\mathbb{Z}^c},$$ where bkm: → are nuclear, $${\left\| \right\|_{{\mathfrak{S}_1}}} \le {\beta _m},\quad k,m $$\left\|\cdot\right\|{_{{\mathfrak{S}_1}}}$$ norm, β ∈ l1(ℤc,ℂ) or l1,g(ℤc,ℂ), g an appropriate weight on ℤc. It established that + invertible, then inverse (1 T)−1 has T1, T1 also nuclear. This result refined for case of operators Lp (ℝc,ℂ).