Compact-like operators between non-archimedean normed spaces

The Fredholm theory for compact operators on a non-archimedean Banach space E, as recently developed by W. Schikhof, does not work if the hypothesis of the completeness of E is dropped. This observation led the authors to introduce two new ideals of operators between non-archimedean normed spaces which, in the case of Banach spaces coincide with the ideal of the compact operators. They also investigate in various ways the possible equality of the three operator ideals.