Monotone Operators Without Enlargements

  • Jonathan M. Borwein
  • Regina S. Burachik
  • Liangjin YaoEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 50)


Enlargements have proven to be useful tools for studying maximally monotone mappings. It is therefore natural to ask in which cases the enlargement does not change the original mapping. Svaiter has recently characterized non-enlargeable operators in reflexive Banach spaces and has also given some partial results in the nonreflexive case. In the present paper, we provide another characterization of non-enlargeable operators in nonreflexive Banach spaces under a closedness assumption on the graph. Furthermore, and still for general Banach spaces, we present a new proof of the maximality of the sum of two maximally monotone linear relations. We also present a new proof of the maximality of the sum of a maximally monotone linear relation and a normal cone operator when the domain of the linear relation intersects the interior of the domain of the normal cone.

Key words

Adjoint Fenchel conjugate Fitzpatrick function Linear relation Maximally monotone operator Monotone operator Multifunction Normal cone operator Non-enlargeable operator Operator of type (FPV) Partial inf-convolution Set-valued operator 

Mathematics Subject Classifications (2010)

47A06 47H05 47B65 47N10 90C25 



The authors thank Dr. Heinz Bauschke and Dr. Xianfu Wang for their valuable discussions and comments. The authors thank an anonymous referee for his/her careful reading and pertinent comments. Jonathan Borwein was partially supported by the Australian Research Council. The third author thanks CARMA at the University of Newcastle and the School of Mathematics and Statistics of University of South Australia for the support of his visit to Australia, which started this research.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Jonathan M. Borwein
    • 1
    • 2
  • Regina S. Burachik
    • 3
  • Liangjin Yao
    • 1
    Email author
  1. 1.CARMAUniversity of NewcastleNewcastleAustralia
  2. 2.King Abdulaziz UniversityJeddahSaudi Arabia
  3. 3.School of Mathematics and StatisticsUniversity of South AustraliaMawson LakesAustralia

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