Positive operators on a free Banach space over the complex Levi-Civita field

Abstract

Let C be the complex Levi-Civita field and let c₀(C) or, simply, c₀ denote the space of all null sequences of elements of C. A non-Archimedean norm is defined naturally on c₀ with respect to which c₀ is a Banach space. In this paper, we study the properties of positive operators on c₀ which are similar to those of positive operators in classical functional analysis; however the proofs of many of the results are nonclassical. Then we use our study of positive operators to introduce a partial order on the set of compact and self-adjoint operators on c₀ and study the properties of that partial order.