Banach's closed range theorem and Fredholm alternative theorem in non- Archimedean Banach spaces (Q797114)

Let E and F be non-Archimedean Banach spaces. A linear operator T of E to F is called completely continuous if for any bounded sequence of vectors \(\{x_ n\}\) in E, the sequence \(\{T(x_ n)\}\) contains a convergent subsequence. However, if there exists a nontrivial completely continuous linear map, then K must be locally compact. \textit{A. van Rooij} [Non- Archimedean functional analysis (1978; Zbl 0396.46061)] has extended this definition to a compact operator. The author has a Fredholm alternative theorem to the compact operator. This is the extended result to the theorem of \textit{L. Narici}, \textit{E. Beckenstein} and \textit{G. Bachman} [Functional analysis and valuation theory (1971; Zbl 0218.46004) p. 91]. Further this paper contains also a new proof of the Banach's closed range theorem in non-Archimedean Banach spaces.