Compact perturbations of \(p\)-adic operators with finite codimensional range (Q2751744)

Let \(X,Y\) be non-archimedean (n.a.) normed spaces over a complete n.a. valued field \(\mathbb K\) [see \textit{A. C. M. van Rooij}, ``Non-Archimedean Functional Analysis'', New York (1978; Zbl 0396.46061)]. A subset \(A\) of \(X\) is called a compactoid if for every \(\epsilon > 0\) there exists a finite set \(Z\subset X\) such that \(A\subset \)co\((Z) + B_X\), where \(B_X\) denotes the closed unit ball of \(X\). A continuous linear operator \(T\in L(X,Y)\) is called compact if \(T(B_X)\) is a compactoid in \(Y\). The operator \(T\in L(X,Y)\) is called semi-Fredholm, denoted as \(T\in \Phi_-(X,Y), \) provided his range \(R(T)\) is finite codimensional: \(\delta (T):= \dim(Y/R(T)) <\infty\). It follows that the range \(R(T)\) is closed. The class \(\Phi_+\) is defined as formed by the operators \(T\) with closed range and finite dimensional kernel. The author studies the stability of various classes of operators, including semi-Fredholm, under perturbations. For instance, if \(T\in \Phi_-(X,Y)\) and \(F \in L(X,Y) \) has finite dimensional range then \(T+F \in \Phi_-(X,Y)\) and \(\delta(T+F) \leq \delta (T) +\dim R(F)\) (Th. 3.3). If \(T\in L(X,Y)\) is surjective and \(S\in L(X,Y)\) satisfies \(\|S\|< \|\widehat T^{-1}\|^{-1}\), where \(\widehat T: X/\ker T\to Y\), then \(T+S\) is surjective too (Cor. 4.3). If \(T\in \Phi_-(X,Y)\) and \(S\) is compact then \(T+S \in \Phi_-(X,Y)\) (Th. 5.3).NEWLINENEWLINEFor the entire collection see [Zbl 0969.00058].