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Maximally monotone operators with ranges whose closures are not convex and an answer to a recent question by Stephen Simons


Authors: Heinz H. Bauschke, Walaa M. Moursi and Xianfu Wang
Journal: Proc. Amer. Math. Soc. 148 (2020), 2035-2044
MSC (2010): Primary 47H05; Secondary 46B20, 47A05
DOI: https://doi.org/10.1090/proc/14859
Published electronically: January 13, 2020
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Abstract: In his recent Proceedings of the AMS paper, ``Gossez's skew linear map and its pathological maximally monotone multifunctions'', Stephen Simons proved that the closure of the range of the sum of the Gossez operator and a multiple of the duality map is not convex whenever the scalar is between 0 and $ 4$. The problem of the convexity of that range when the scalar is equal to $ 4$ was explicitly stated. In this paper, we answer this question in the negative for any scalar greater than or equal to $ 4$. We derive this result from an abstract framework that allows us to also obtain a corresponding result for the Fitzpatrick-Phelps integral operator.


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Additional Information

Heinz H. Bauschke
Affiliation: Department of Mathematics, University of British Columbia, Kelowna, British Columbia V1V 1V7, Canada
Email: heinz.bauschke@ubc.ca

Walaa M. Moursi
Affiliation: Department of Electrical Engineering, Stanford University, 350 Serra Mall, Stanford, California 94305
Email: wmoursi@stanford.edu

Xianfu Wang
Affiliation: Department of Mathematics, University of British Columbia, Kelowna, British Columbia V1V 1V7, Canada
Email: shawn.wang@ubc.ca

DOI: https://doi.org/10.1090/proc/14859
Keywords: Duality map, Fitzpatrick-Phelps operator, Gossez operator, maximally monotone operator, range, skew operator
Received by editor(s): May 9, 2019
Received by editor(s) in revised form: July 16, 2019, July 18, 2019, July 19, 2019, September 3, 2019, and September 6, 2019
Published electronically: January 13, 2020
Additional Notes: The research of the first and third authors was partially supported by Discovery Grants of the Natural Sciences and Engineering Research Council of Canada
The research of the second author was partially supported by the Natural Sciences and Engineering Research Council of Canada Postdoctoral Fellowship
Communicated by: Stephen Dilworth
Article copyright: © Copyright 2020 American Mathematical Society