Abstract

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 concepts of "rLr_L-density" and its specialization "quasidensity": the closed quasidense monotone multifunctions from a Banach space into its dual form a (generally) strict subset of the maximally monotone ones, though all surjective maximally monotone and all maximally monotone multifunctions on a reflexive space are quasidense. We give a sum theorem and a parallel sum theorem for closed monotone quasidense multifunctions under very general constraint conditions. That is to say, quasidensity obeys very nice calculus rules. We give a short proof that the subdifferential of a proper convex lower semicontinuous function on a Banach space is quasidense, and deduce generalizations of the Brezis-Browder theorem on linear relations to non reflexive Banach spaces. We also prove that any closed monotone quasidense multifunction has a number of other very desirable properties.

Author information

Contact details are reproduced from the original publication and may be historical.

Stephen Simons

Dept. of Mathematics, University of California, Santa Barbara, CA 93106-3080, U.S.A.

stesim38@gmail.com

Keywords

  • Banach SN space
  • L-positive set
  • r-L-density
  • quasidensity
  • multifunction
  • maximal monotonicity
  • sum theorem
  • subdifferential
  • negative alignment
  • monotone linear relation
  • Brezis-Browder theorem

Mathematics Subject Classification

47H05; 47N10, 52A41, 46A20

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Suggested citation

S. Simons. “"Densities" and Maximal Monotonicity.” Journal of Convex Analysis 23 (2016), No. 4, 1017–1050. https://doi.org/10.68381/jca23038

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Published by Heldermann Verlag, 2016. Rights now held by Banach Press.