Construction of pathological maximally monotone operators on non-reflexive Banach spaces

TL;DR

This paper constructs specific maximally monotone operators in non-reflexive Banach spaces that lack certain properties, revealing limitations in the behavior of BC-functions and answering a longstanding open question.

Contribution

It introduces new examples of non-type (D) maximally monotone operators in non-reflexive Banach spaces, challenging existing assumptions about BC-functions and the structure of such operators.

Findings

Constructed maximally monotone operators not of Gossez's dense-type (D)

Showed partial inf-convolution of BC-functions may not be a BC-function

Demonstrated existence of non type (D) operators in spaces containing James space, c0, or ℓ¹

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 Br{\o}nsted-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 that every Banach space which contains an isomorphic copy of the James space $\mathbf{J}$ or its dual $\mathbf{J}^*$, or $c_0$ or its dual $\ell^1$, admits a non type (D) operator.