On \(p\)-adic Fredholm operators with null index (Q5929777)

Let \(X\), \(Y\) be Banach spaces over a non-Archimedean valued complete field. As in the complex case, a continuous linear operator \(T: X\to Y\) is said to be Fredholm if \(\ker T\) and \(Y/\text{Im} T\) are finite-dimensional. \(T\) has null index if those dimensions are equal. After an introduction on compact perturbations of \(T\), it is proved that for a Fredholm \(T\) with null index and a compact \(K: X\to Y\), bijectivity, surjectivity of \(T+K\) are equivalent. The fact that \(\ker T\) may not be topologically complemented in \(X\) causes derivations from the `classical' theory. The following properties are considered. (a) \(T\) Fredholm, with null index, (b) there is a finite rank operator \(F\) with \(T+F\) is bijective, (c) there is a compact \(K\) with \(T+K\) is bijective. It is shown that (b), (c) are each equivalent to (a) is `\(\ker T\) is complemented''. Several examples of cases where (a) \(\Rightarrow\) (b) does not hold are given.