## 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?*