Maximally monotone operators with ranges whose closures are not convex and an answer to a recent question by Stephen Simons
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- by Heinz H. Bauschke, Walaa M. Moursi and Xianfu Wang PDF
- Proc. Amer. Math. Soc. 148 (2020), 2035-2044 Request permission
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.References
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Additional Information
- Heinz H. Bauschke
- Affiliation: Department of Mathematics, University of British Columbia, Kelowna, British Columbia V1V 1V7, Canada
- MR Author ID: 334652
- Email: heinz.bauschke@ubc.ca
- Walaa M. Moursi
- Affiliation: Department of Electrical Engineering, Stanford University, 350 Serra Mall, Stanford, California 94305
- MR Author ID: 983132
- Email: wmoursi@stanford.edu
- Xianfu Wang
- Affiliation: Department of Mathematics, University of British Columbia, Kelowna, British Columbia V1V 1V7, Canada
- MR Author ID: 601305
- Email: shawn.wang@ubc.ca
- 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
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 2035-2044
- MSC (2010): Primary 47H05; Secondary 46B20, 47A05
- DOI: https://doi.org/10.1090/proc/14859
- MathSciNet review: 4078087