Journal of Convex Analysis, Vol. 7, No. 1, pp. 115-128 (2000)

Regular Maximal Monotone Operators and the Sum Theorem

Andrei Verona and Maria Elena Verona

Department of Mathematics, California State University, Los Angeles, CA 90032, USA, and Department of Mathematics, University of Southern California, Los Angeles, CA 90089-1113, USA,

Abstract: In this note, which is a continuation of [17], we study two classes of maximal monotone operators on general Banach spaces which we call ${\cal C}_0$ (resp. ${\cal C}_1$)-{\mit regular}. All maximal monotone operators on a reflexive Banach space, all subdifferential operators, and all maximal monotone operators with domain the whole space are ${\cal C}_1$-regular and all linear maximal monotone operators are ${\cal C}_0$-regular. We prove that the sum of a ${\cal C}_0$ (or ${\cal C}_1$)-regular maximal monotone operator with a maximal monotone operator which is locally inf bounded and whose domain is closed and convex is again maximal monotone provided that they satisfy a certain "dom-dom" condition. From this result one can obtain most of the known sum theorem type results in general Banach spaces. We also prove a local boundedness type result for pairs of monotone operators.

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