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

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\ensuremath{\mathbf{J}}}$\end{document} or its dual \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\ensuremath{\mathbf{J}}}^{\ast}$\end{document}, or c0 or its dual ℓ1, admits a non type (D) operator. The existence of non type (D) operators in spaces containing ℓ1 or c0 has been proved recently by Bueno and Svaiter.