Wiener's lemma for localized integral operators

In this paper, we introduce two classes of localized integral operators on L2(Rd) with the Wiener class W and the Kurbatov class K of integral operators as their models. We show that those two classes of localized integral operators are pseudo-inverse closed non-unital subalgebra of B2, the Banach algebra of all bounded operators on L2(Rd) with usual operator norm.