Matrix characterizations of Lipschitz operators on Banach spaces over Krull valued fields (Q2381133)

An important research area in \(p\)-adic functional analysis is the theory of normed spaces over a Krull valued field, i.e., a field \(K\) together with a surjective map, called Krull valuation, \(\left| \, \, \right| : K \rightarrow G \cup \{ 0 \}\) (where \(G\) is a linearly ordered Abelian group, written multiplicatively, augmented with a zero element \(0\)) that satisfies the properties of a real-valued non-Archimedean valuation, replacing \([0,\infty)\) by \(G \cup \{ 0 \}\). Similarly, the norms take values in \(X \cup \{0 \}\) (where \(X\) is a \(G\)-module) and satisfy the properties of a real-valued non-Archimedean norm, replacing now \([0,\infty)\) by \(X \cup \{ 0 \}\). The authors are experts in this part of \(p\)-adic functional analysis and belong to the main contributors responsible for its progress in recent years. One of the subjects that they studied deals with operators between normed spaces over \(K\). This paper is the natural continuation of their previous ones on this subject. Let \(E\) be a \(K\)-Banach space with a countable orthogonal base \(e_1, e_2, \dots\). Examples of particular interest of such \(E\), which are also considered in this paper, are the so-called norm Hilbert spaces, i.e., \(K\)-Banach spaces in which each closed subspace has an orthogonal complement. Let \(A: E \rightarrow E\) be a linear operator. \(A\) is called Lipschitz, resp., strictly Lipschitz, if there is a \(g\in G\) such that \(\| A(x) \| \leq g \| x \|\) for all \(x \in E\), resp., \(\| A(x) \| < g \| x \|\) for all nonzero \(x \in E\) (not every Lipschitz operator is strictly Lipschitz; a surprising fact!). \(A\) is called compact (resp., nuclear) if \(A\) is in the closure of the space of continuous operators of finite rank with respect to the Lipschitz norm \(\| \;\| \) (resp., the strictly Lipschitz norm \(\| \;\| ^{\sim}\)) on the corresponding space of operators. The purpose of the present paper is to characterize the above classes of operators in terms of their (infinite) matrices \((a_{mn})\) (\(m,n \in \mathbb N\)) with respect to the given base \(e_1, e_2, \dots\). Of special interest are the ``building blocks'' \(P_{mn}\) given by the formula \(P_{mn}(e_k) = \delta_{kn} e_m\) (\(k\in\mathbb N\)). It is proved that for a linear operator \(A: E \rightarrow E\), the following are equivalent. {\parindent=6mm \begin{itemize}\item[(i)]\(A\) is compact (resp., \(A\) is Lipschitz). \item[(ii)]\(\lim_m \| a_{mn} P_{mn} \| =0\) uniformly in \(n\) (resp., for each \(n\), \(\lim_m \| a_{mn} P_{mn} \| =0\) and \((m,n) \mapsto \| a_{mn} P_{mn}\| \) is bounded above). \end{itemize}} By replacing in (ii) \(\| \;\| \) by \(\| \;\| ^{\sim}\), we get in (i), ``\(A\) is nuclear (resp., \(A\) is strictly Lipschitz)''. The above result is not only interesting in its own right but can generate concrete (counter-)examples of operators having certain desired properties, and provides useful tests to decide whether an operator is Lipschitz or not.