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Journal of Convex Analysis, Vol. 7, No. 1, pp. 115-128 (2000)
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Regular Maximal Monotone Operators and the Sum Theorem

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Andrei Verona and Maria Elena Verona

Department of Mathematics, California State University, Los Angeles, CA 90032, USA, averona@calstatela.edu and Department of Mathematics, University of Southern California, Los Angeles, CA 90089-1113, USA, verona@math.usc.edu

**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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