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A \(p\)-adic version of Hilbert–Schmidt operators and applications. (English) Zbl 1077.47061
Hilbert–Schmidt operators form one of the most important classes of operators between real or complex Hilbert spaces, because of their influence in the applications. In this paper, the authors initiate the study of a \(p\)-adic version of Hilbert–Schmidt operators on what they call \(p\)-adic Hilbert spaces. The purpose is to prove some of their basic properties (existence of adjoint, ideal structure, complete continuity, trace). Unfortunately, the paper has an intense archimedean character. Most of the results and proofs are essentially adaptations of their classical analogues. The same happens with some of the central concepts. This is the case of the definition of Hilbert–Schmidt operator in terms of a sequence that is in the space \(\ell^{2}\) (which does not belong to the non-archimedean world) and of the definition of Hilbert–Schmidt norm (which does not satisfy the strong triangle inequality required for any non-archimedean norm).
The reviewer believes that \(p\)-adic Hilbert–Schmidt operators promise to be an interesting research subject in \(p\)-adic functional analysis and its applications. Nevertheless, I recommend to the authors to develop their future \(p\)-adic investigation with the aim of obtaining results with a typically non-archimedean nature. Presumably, this process implies to introduce new concepts that are more appropriate to the \(p\)-adic context. This is the way of proceeding to do “real” \(p\)-adic mathematics.

47S10 Operator theory over fields other than \(\mathbb{R}\), \(\mathbb{C}\) or the quaternions; non-Archimedean operator theory
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis