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

摘要

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 IK and the abelian group G. In particular, we consider the Banach space of countable type co consisting of all null sequences of K((G)), equipped with the supremumnorm ||?||∞and we define a natural inner product on co which induces the norm of co. Then we present characterizations of normal projections and of compact and self-adjoint operators on co. 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 co 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 co and use that to introduce a partial order on the set of compact and self-adjoint operators on c_0.