Perturbation theory of \(p\)-adic Fredholm and semi-Fredholm operators.
Let \(X,Y\) be a non-archimedean (n.a.) Banach spaces over a n.a. valued field \(\mathbb K\,\) [see \textit{A. C. M. van Rooij,} Non-Archimedean functional analysis (1978; Zbl 0396.46061)]. A continuous linear operator \(T:X\to Y\) is called semi-Fredholm\((-)\) provided that \(\eta(T):= \dim \ker(T) < \infty\) and the range \(R(T)\) is closed in \(Y,\) and semi-Fredholm(+) provided that \(\,\delta(T) : = \) dim \(Y/R(T) < \infty\) (in this case, \(R(T)\) is necessarily closed). Denote by \(\Phi_ -(X,Y)\) and \(\Phi_ +(X,Y)\) the corresponding sets of operators and let \(\Phi (X,Y) = \Phi_ -(X,Y)\cap \Phi_ +(X,Y)\) be the set of Fredholm operators from \(X\) to \(Y\). The quantity \(\chi (T) := \eta(T) - \delta(T)\) is called the index of the operator \(T\). The paper under review is concerned with stability properties of these classes of operators with respect to small or compact perturbations, completing the results obtained by \textit{J. Araujo, C. Pérez-García} and \textit{S. Vega} [Compos. Math. 118, 291--303 (1999; Zbl 1038.47047)] and \textit{S. Vega} [Lect. Notes Pure Appl. Math. 222, 301--307 (2001; Zbl 0993.47044)]. At the same time, the obtained results form a non-archimedean counterpart of some well-known classical results in operator theory. Since a finite-dimensional subspace of a non-archimedean Banach \(X\) space is not necessarily topologically complemented in \(X\), the authors have to work with orthogonal complements corresponding to a notion of non-archimedean orthogonality called \(t\)-orthogonality and meaning that for \(t\in (0,1]\) the sequence \((e_ i)\) satisfies the inequalities \(t\max\{\| \lambda _ i e_ i\| : 1\leq i\leq k\} \leq \| \sum_{i=1}^ k\lambda_ i e_ i\|\) for all \(k\in \mathbb N\) (or \(k\) not exceeding the length of the sequence \((e_ i)\)) and all \(\lambda_ 1,\dots,\lambda_ k\in \mathbb K.\)
scientific article; zbMATH DE number 2124633
scientific article