Common extensions for linear operators
Volume 95 / 2011
Abstract
The main meaning of the common extension for two linear operators is the following: given two vector subspaces $G_{1}$ and $G_{2}$ in a vector space (respectively an ordered vector space) $E$, a Dedekind complete ordered vector space $F$ and two (positive) linear operators $ T_{1}:G_{1}\rightarrow F$, $T_{2}:G_{2}\rightarrow F$, when does a (positive) linear common extension $L$ of$\ T_{1}$, $T_{2}$ exist?
First, $L$ will be defined on $\mathop{\rm span}( G_{1}\cup G_{2}) $. 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 $T_{1}$, $T_{2}$ 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 $G_{1}$, $G_{2}$ there are some arbitrary nonempty sets, instead of $T_{1}$, $T_{2}$ there are two arbitrary maps $f_{1}$, $ f_{2}$, and, in addition, we are given two more maps $g_{1}:G_{1}\rightarrow E,\ g_{2}:G_{2}\rightarrow E$ and a sublinear operator $S:E\rightarrow F$. In this case we ask: When is it possible to obtain a linear operator $L:E\rightarrow F$, dominated by $S$ and related to the maps $f_{1}$, $f_{2}$, $g_{1}$, $g_{2}$ 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.
