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Set-Valued and Variational Analysis

, Volume 20, Issue 3, pp 387–415 | Cite as

Construction of Pathological Maximally Monotone Operators on Non-reflexive Banach Spaces

  • Heinz H. Bauschke
  • Jonathan M. BorweinEmail author
  • Xianfu Wang
  • Liangjin Yao
Article

Abstract

In this paper, we construct maximally monotone operators that are not of Gossez’s dense-type (D) in many nonreflexive spaces. Many of these operators also fail to possess the Brønsted-Rockafellar (BR) property. Using these operators, we show that the partial inf-convolution of two BC–functions will not always be a BC–function. This provides a negative answer to a challenging question posed by Stephen Simons. Among other consequences, we deduce—in a uniform fashion—that every Banach space which contains an isomorphic copy of the James space \({\ensuremath{\mathbf{J}}}\) or its dual \({\ensuremath{\mathbf{J}}}^{\ast}\), or c 0 or its dual ℓ1, admits a non type (D) operator. The existence of non type (D) operators in spaces containing ℓ1 or c 0 has been proved recently by Bueno and Svaiter.

Keywords

Adjoint BC–function Fitzpatrick function James space Linear relation Maximally monotone operator Monotone operator Multifunction Operator of type (BR) Operator of type (D) Operator of type (NI) Partial inf-convolution Schauder basis Set-valued operator Shrinking basis Skew operator Space of type (D) Subdifferential operator Uniqueness of extensions 

Mathematics Subject Classifications (2010)

Primary 47A06 47H05; Secondary 47B65 47N10  90C25 

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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Heinz H. Bauschke
    • 1
  • Jonathan M. Borwein
    • 2
    • 3
    Email author
  • Xianfu Wang
    • 1
  • Liangjin Yao
    • 2
  1. 1.Mathematics, Irving K. Barber SchoolUniversity of British ColumbiaKelownaCanada
  2. 2.CARMAUniversity of NewcastleNewcastleAustralia
  3. 3.King Abdulaziz UniversityJeddahSaudi Arabia

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