The main meaning of the common extension for two linear operators is the following: given two vector subspaces G₁ and G₂ in a vector space (respectively an ordered vector space) E, a Dedekind complete ordered vector space F and two (positive) linear operators T₁: G₁ → F, T₂: G₂ → F, when does a (positive) linear common extension L of T₁, T₂ exist?
First, L will be defined on span(G₁ ∪ G₂). In other results, formulated in the line of the Hahn-Banach extension theorem, the common extension L will be defined on the whole space E, by requiring the majorization of T₁, T₂ by a (monotone) sublinear operator. Note that our first Hahn-Banach common extension results were proved by using two results formulated in the line of the Mazur-Orlicz theorem. Actually, for the first of these last mentioned results, we extend the name common extension to the case when E is without order structure, instead of G₁, G₂ there are some arbitrary nonempty sets, instead of T₁, T₂ there are two arbitrary maps f₁, f₂, and, in addition, we are given two more maps g₁: G₁ → E, g₂: G₂ → E and a sublinear operator S: E → F. In this case we ask: When is it possible to obtain a linear operator L: E → F, dominated by S and related to the maps f₁, f₂, g₁, g₂ by some inequalities?
To extend positive linear operators between ordered vector spaces, some authors (Z. Lipecki, R. Cristescu and myself) have used a procedure which includes the introduction of an additional set and a corresponding map. Inspired by this technique, in this paper we also solve some common positive extensions problems by using an additional set.