$r_L$-density and maximal monotonicity

Stephen Simons

Published 2014-07-04, updated 2014-10-27Version 2

We discuss "Banach SN spaces", which include Hilbert spaces, negative Hilbert spaces, and the product of any real Banach space with its dual. We introduce "L-positive" sets, which generalize monotone multifunctions from a Banach space into its dual. We introduce the concept of $r_L$-density: the closed $r_L$-dense monotone multifunctions from a Banach space into its dual form a strict subset of the maximally monotone ones. We give two short proofs that the subdifferential of a proper convex lower semicontinuous function on a Banach space is $r_L$-dense. We give sum theorems for closed monotone $r_L$-dense multifunctions under very general constraint conditions, and we prove that any closed monotone $r_L$-dense multifunction is of type (ANA), strongly maximally monotone, of type (FPV), of type (FP), of type (NI), and has a number of other very desirable properties. In the (FP) and (NI) case the converse results are true.