On an Operator Theory on a Banach Space of Countable Type over a Hahn Field

Abstract

This paper is a survey of the results in Aguayo et al. (J Math Phys 54(2), 2013; Indag Math (N.S.) 26(1):191–205, 2015; p-Adic Num Ultrametr Anal Appl 9(2):122–137, 2017) but generalized to the case when the complex Levi-Civita field C is replaced by a Hahn field K((G) ) for particular choices of the field K and the abelian group G. In particular, we consider the Banach space of countable type c 0 consisting of all null sequences of K((G) ), equipped with the supremum norm ∥⋅∥ ∞ and we define a natural inner product on c 0 which induces the norm of c 0 . Then we present characterizations of normal projections and of compact and self-adjoint operators on c 0 . As a new result in this paper, we apply the Hahn–Banach theorem to show the existence of the dual operator of a given continuous linear operator on c 0 and to show that the dual operator and the adjoint operator coincide. We present some B ∗ -algebras of operators, including those mentioned above, then we define an inner product on such algebras which induces the usual norm of operators. Finally, we present a study of positive operators on c 0 and use that to introduce a partial order on the set of compact and self-adjoint operators on c 0 .