Spectral Theory of Linear Operators on Non-Archimedean Quasi-Banach Spaces

In this paper, we introduce the notion of free non-Archimedean quasi-Banach spaces. We prove some results on non-Archimedean quasi-Banach spaces and we give some examples of such spaces. The uniform boundedness principle, the closed graph theorem, the Banach's open mapping theorem and the bounded inverse theorem for non-Archimedean quasi-Banach spaces are proved. Furthermore, we define the notions of closed linear operators, bounded below operators, invertible operators, r-spectral operators, finite rank operators and completely continuous operators on non-Archimedean quasi-Banach spaces and we establish some results about them. The spectral theory of bounded linear operators is studied. Finally, the quasi-norm convergence, the quasi-pointwise convergence and the quasi nu-convergence are introduced and studied. Several examples are provided.