A note on generalized invertibility and invariants of infinite matrices

The classes of band-dominated operators and the subclass of operators in the Wiener algebra $${\mathcal {W}}$$ W are known to be inverse closed. This paper studies and extends partially known results of that type for one-sided and generalized invertibility. Furthermore, for the operators in the Wiener algebra $${\mathcal {W}}$$ W invertibility, the Fredholm property and the Fredholm index are known to be independent of the underlying space $$l^p$$ l p , $$1\le p\le \infty $$ 1 ≤ p ≤ ∞ . Here this is completed by the observation that even the kernel and a suitable direct complement of the range as well as generalized inverses of operators in $${\mathcal {W}}$$ W are invariant w.r.t. p.