Abstract

We study a question posed by Stephen Simons in his 2008 monograph involving "bigger conjugate" (BC) functions and the partial infimal convolution. As Simons demonstrated in his monograph, these function have been crucial to the understanding and advancement of the state-of-the-art of harder problems in monotone operator theory, especially the sum problem. In this paper, we provide some tools for further analysis of BC-functions which allow us to answer Simons' problem in the negative. We are also able to refute a similar but much harder conjecture which would have generalized a classical result of Brézis, Crandall and Pazy. Our work also reinforces the importance of understanding unbounded skew linear relations to construct monotone operators with unexpected properties.

Author information

Contact details are reproduced from the original publication and may be historical.

Heinz H. Bauschke

Dept. of Mathematics, Irving K. Barber School, University of British Columbia, Kelowna, B.C. V1V 1V7, Canada

heinz.bauschke@ubc.ca

Xianfu Wang

Dept. of Mathematics, Irving K. Barber School, University of British Columbia, Kelowna, B.C. V1V 1V7, Canada

shawn.wang@ubc.ca

Liangjin Yao

Dept. of Mathematics, Irving K. Barber School, University of British Columbia, Kelowna, B.C. V1V 1V7, Canada

ljinyao@interchange.ubc.ca

Keywords

  • Adjoint
  • BC-function
  • Fenchel conjugate
  • Fitzpatrick function
  • linear relation
  • maximally monotone operator
  • monotone operator
  • multifunction
  • normal cone operator
  • partial infimal convolution

Mathematics Subject Classification

47A06, 47H05; 47B65, 47N10, 90C25

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Suggested citation

H. H. Bauschke, J. M. Borwein, X. Wang, L. Yao. “Monotone Operators and "Bigger Conjugate" Functions.” Journal of Convex Analysis 20 (2013), No. 1, 143–155. https://doi.org/10.68381/jca20009

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Published by Heldermann Verlag, 2013. Rights now held by Banach Press LLC.