Subdifferentials of convex operators

SUBDIFFERENTIALS S.

S.

OF

CONVEX

OPERATORS

Kutateladze

UDC

513.88

0% Let X be a v e c t o r space, Y a K - s p a c e , and ~ a weakly o r d e r w i s e bounded set in space L(X, Y) of linear o p e r a t o r s , acting f r o m X into Y. This means that the sublinear o p e r a t o r

P~:x -~ sup {Ax:A~t} is defined in the entire space X.

The set cop (f~)--~ {A~L(X, Y) :Ax~P~x

(x~X)}

is called t:he support hull of the set ~. Clearly, ~l~cop (0/)if and only if ~ is the support set of an e v e r y w h e r e defined sublinear o p e r a t o r Ps, i . e . , is its subdifferential (at zero) O(P ~). A principal topic in theory of sublinear (and a r b i t r a r y convex) o p e r a t o r s is the problem of explicitly d e s c r i b i n g the natural Minkowski duality, i . e . , the problem of i n t e r i o r c h a r a c t e r i z a t i o n of support hulls. Closely connected with this is the problem of evaluating the subdifferentiais of various composite convex o p e r a t o r s . These topics have been examined in a number of a r t i c l e s (see, e . g . , [1-5]). Yet the list of evaluated subdifferentials is quite small. Even less is known about the factoring of the o p e r a t o r ~-)-cop (~)of c l o s u r e in M o o r e ' s sense, i . e . , about its r e p r e s e n t a t i o n as the composition of a s i m i l a r algebraic o p e r a t o r and a c l o s u r e in the sense of some topology. Our p r e s e n t approach r e s t s on the fact that, to examine any sublinear (or convex) o p e r a t o r s , we need to d i s c o v e r how to handle just a "single" canonical o p e r a t o r , in t e r m s of which the action of any o p e r a t o r can be t r a n s m i t t e d , in such a way that the residue is linear (or affine). This canonical o p e r a t o r is defined by a set of p r o j e c t o r s onto a prime r e g u l a r subspace, the aim being to construct, e.g., the required factorings, with the aid of these p r o j e c t o r s . As examples of the application of our approach, we evaluate the subdifferentials of ~he m a x i m u m of convex o p e r a t o r s ; and we p e r f o r m natural factorings of the support hull o p e r a t o r in the case of d i s c r e t e K - s p a c e s , and for "compactly generated" sets in the case of o p e r a t o r s , acting on the b a s e s of K - s p a c e s of bounded e l e ments. To simplify (however defined) applications, such continuous, in the 1 ~ LetYbe h~g ~ (g)A~. We

the treatment, we primarily consider constraints are imposed on the linear a restriction is not in fact essential, sense in which the initial operator is

the vector version of the theory, i.e., no continuity operators employed. From the point of view of since support operators are, as a rule, automatically continuous.

a K-space, ~ a set, andA~:Y-~Ygthe imbedding of Y lathe diagonal of the space Yg, i.e.~ denote by (Y g)~ the basis of the K-space Yg defined by the relation

Notice that, given any subset ~' in ~/, the space (Y ~')~ admits natural mapping with a component Pr~,/~(Y~ ) ~ ) , where the p r o j e c t o r Pr~, is given by (Pr_~, ((gA)A~))A' = [0,

A' ~ ~'.

In the space (Y-~)~, the canonical sublinear o p e r a t o r eg: (Y~) ~ --,- Y acts in accordance with the rule

e,a: (g.~)-~-~ st, p {g~:A~g}. In the case when ~/ is a weakly o r d e r w i s e bounded subset in the space L(X, Y), there a r i s e s the natural linear o p e r a t o r [~] : X - ~ (Y~)~, given by the relation [~/] :x--~- ( A x ) ~ . 1977.

T r a n s l a t e d f r o m Sibirskii Matematicheskii Zhurnal, Vol. 18, No. 5, pp. 1057-1064, S e p t e m b e r - O c t o b e r , Original a r t i c l e submitted June 3, 1976.

0037-4466/77/1805-0747507.50

9 1978 Plenum

Publishing

Corporation

747

We introduce some notation. We denote by Iy the identity mapping of the space Y into itseff. By ~qf+(X, Y), where X and Y a r e o r d e r e d v e c t o r spaces, we denote as usual the set of positive linear o p e r a t o r s

in L(X, Y). The point of introducing the above constructions can be seen f r o m the following propositions. P r o p o s i t i o n 1.

Let P :X - - Y be a sublinear o p e r a t o r , where O ( P ) = c o p (~). Then P-----e~to [~].

P r o p o s i t i o n 2.

We have the r e p r e s e n t a t i o n s o(~) = {~+((Y~)~,

Y):~~

cop ( ~ ) = a ( ~ )

P r o p o s i t i o n 3.

A~=I~};

~ [~].

If Z is a K - s p a c e and P :Y --- Z is an i n c r e a s i n g sublinear o p e r a t o r , then O(P o e~) ---- {A~-q~+ ((Y~)|

Z) : A ~ A ~---a(p)}.

T h e s e propositions can be proved by d i r e c t calculation. LetAy~- 0 a n d 0 - < y ~ z, t h e n a n o p e r a t o r 0 - ~ Iy exists, f o r which y = flz, s i n c e the i n t e r v a l [0, z] is the s a m e a s the s e t 8(P0)z, w h e r e P 0 : z --~ z+. L e t ~ - i :c~(Y} --* Y b e the o p e r a t o r , i n v e r s e to ~ . If a y > 0, a n d it is not t r u e that y > 0, then we have P r y < 0 for s o m e p r o j e c t o r P r . H e n c e c ~ P r y