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Unbounded linear monotone operators on nonreflexive Banach spaces

Summary: Unbounded linear monotone operators on
nonreflexive Banach spaces
R. R. Phelps and S. Simons
1. Introduction.
Many of the most useful results concerning maximal monotone set--valued operators are
only valid in reflexive Banach spaces. In an effort to extend some of these results to
nonreflexive spaces, various authors have introduced certain natural subclasses of maxi­
mal monotone operators (subclasses which are identical with the entire class of maximal
monotone operators in reflexive spaces). The precise relationships between the various new
classes of operators have remained murky, although it is very clear in a particular case:
Bauschke and Borwein [2,3] have shown that all of these notions coincide for bounded linear
monotone operators. In this note, we consider the next simplest case, namely, unbounded
linear monotone operators. It may be true that the Bauschke--Borwein characterization
remains valid for unbounded monotone operators; Theorem 6.4 below strongly suggests
that such is the case. (The simplifications needed to prove these facts about unbounded
operators lead to much simpler proofs of the corresponding portions of [3, Theorem 4.1]
--- see Theorem 8.1.)
After some basics in Sections 2 and 3, in Section 4 we look at adjoint operators and
in Section 5 we characterize those linear operators which are subdifferentials of proper
lower semicontinuous convex functions. Section 6 is devoted to those operators which


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics