On sum of range sets of sum of two maximal monotone operators

Document Type : Research Paper


Department of Mathematics, National Institute of Technology, Rourkela, India


In the setting of non-reflexive spaces (Grothendieck Banach spaces), we establish
(1) $\overline{ran (A+B)}=\overline{ran A+ran B}$
(2) int (ran (A+B))=int(ran A+ran B).
with the assumption that A is a maximal monotone operator and B is a single-valued maximal monotone operator such that A+B is ultramaximally monotone. Conditions (1) and (2) are known as Br$\acute{e}$zis-Haraux conditions.