Construction of Pathological Maximally Monotone Operators on Non-reflexive Banach Spaces |
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Authors: | Heinz H Bauschke Jonathan M Borwein Xianfu Wang Liangjin Yao |
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Institution: | 1.Mathematics, Irving K. Barber School,University of British Columbia,Kelowna,Canada;2.CARMA,University of Newcastle,Newcastle,Australia;3.King Abdulaziz University,Jeddah,Saudi Arabia |
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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 ?1, admits a non type (D) operator. The existence of non type (D) operators in spaces containing ?1 or c 0 has been proved recently by Bueno and Svaiter. |
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