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

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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.

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Authors and Affiliations

  1. Mathematics, Irving K. Barber School, University of British Columbia, Kelowna, B.C. V1V 1V7, Canada

    Heinz H. Bauschke & Xianfu Wang

  2. CARMA, University of Newcastle, Newcastle, NSW, 2308, Australia

    Jonathan M. Borwein & Liangjin Yao

  3. King Abdulaziz University, Jeddah, Saudi Arabia

    Jonathan M. Borwein

Authors
  1. Heinz H. Bauschke
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  2. Jonathan M. Borwein
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  3. Xianfu Wang
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  4. Liangjin Yao
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Correspondence to Jonathan M. Borwein.

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Bauschke, H.H., Borwein, J.M., Wang, X. et al. Construction of Pathological Maximally Monotone Operators on Non-reflexive Banach Spaces. Set-Valued Var. Anal 20, 387–415 (2012). https://doi.org/10.1007/s11228-012-0209-0

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