Maximal-Valued Extensions of Positive Operators

✍ Scribed by Zbigniew Lipecki

• Publisher: John Wiley and Sons
• Year: 1984
• Tongue: English
• Weight: 151 KB
• Volume: 117
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.3211170104

✦ Synopsis

We itre concerned with existence of extensions of positive linear operators be-I t v w i i ordered vector spaces which take maximal possible values on a given set of \wit ors. We eatablish a criterion (Theorem) which partially generalizes a similar twiilt of [2] about positive additive set functions (cf. Corollary 1 below). This cri-I i w i o i i also applies to the caae of operators on C,(Q), the vector lattice of real-valued Ilciiinded continuous functions on a normal topological space D (Corollary 2).

Let us note that there exists a related but weaker maximality condition which 4 wiginates from CHOQUET theory. That oondition has been studied, among other rlitlhors, by ANDENAES [l]. In contrast with [l], all our proof8 are effective.

Throughout Y stands for an order complete vector lattice over the reals R. Let X Iio a majorizing (i.e., cofinal) suhspace of an ordered leal vector space X. Given t i I mitive linear operator T : 2 -r Y, we put T,(s)=inf ( T ( z ) : X S Z C Z } 111r r ~l l s c X . (This notation follows [3].) As emily seen, T , ib: subadditive and posi-I ivdy homogeneous. Moreover, if 8 : X -Y is a positive linear operator which ex-I~*INIR T, then S ( s ) z T , ( x ) for all x E X . 'Sheorem. Let T : Z -Y be a positive linear operator a,nd let V & X . Then T extr*rrrls to a (unique) podive l i w r operator S : lin (2 U V) -1' with S(v) = T , ( v ) for all I ' I I' if a d only it / o r all syqterm v , , ..., v , ~ V of different elements. Moreover, S ;Q an ecctreme extension 111 7'. J'roof. The "only if" part IS clear. To prove the "if" part, we fimt show