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Journal of Convex Analysis 07 (2000), No. 1, 115--128
Copyright Heldermann Verlag 2000

Regular Maximal Monotone Operators and the Sum Theorem

Andrei Verona
Dept. of Mathematics, California State University, Los Angeles, CA 90032, U.S.A.

Maria Elena Verona
Dept. of Mathematics, University of Southern California, Los Angeles, CA 90089-1113, U.S.A.

In this note, which is a continuation of a previous paper of the authors [Set-Valued Analysis 6 (1998) 302-312], we study two classes of maximal monotone operators on general Banach spaces which we call C0 (resp. C1)-regular. All maximal monotone operators on a reflexive Banach space, all subdifferential operators, and all maximal monotone operators with domain the whole space are C1-regular and all linear maximal monotone operators are C0-regular. We prove that the sum of a C0 (or C1)-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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