Examples of discontinuous maximal monotone linear operators and the solution to a recent problem posed by BF Svaiter

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:370

Article

Bauschke, Heinz H.¹  Wang, Xianfu¹  Yao, Liangjin¹ 

[1] UBC Okanagan, Irving K Barber Sch, Kelowna, BC V1V 1V7, Canada

关键词: Adjoint operator;    Fitzpatrick function;    Fenchel conjugate;    Linear relation;    Maximal monotone operator;    Multifunction;    Monotone operator;    Skew operator;    Unbounded linear monotone operator;   

DOI  :  10.1016/j.jmaa.2010.04.029

来源: Elsevier

【 摘 要 】

In this paper, we give two explicit examples of unbounded linear maximal monotone operators. The first unbounded linear maximal monotone operator S on l(2) is skew. We show its domain is a proper subset of the domain of its adjoint S*. and -S* is not maximal monotone. This gives a negative answer to a recent question posed by Svaiter. The second unbounded linear maximal monotone operator is the inverse Volterra operator T on L(2)[0, 1]. We compare the domain of T with the domain of its adjoint T* and show that the skew part of T admits two distinct linear maximal monotone skew extensions. These unbounded linear maximal monotone operators show that the constraint qualification for the maximality of the sum of maximal monotone operators cannot be significantly weakened, and they are simpler than the example given by Phelps-Simons. Interesting consequences on Fitzpatrick functions for sums of two maximal monotone operators are also given. (C) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

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