Decompositions and representations of monotone operators with linear graphs

• Main Author: Yao, Liangjin
• Language: English
• Published: University of British Columbia 2008
• Subjects: Maximal montone operator; Anixymmetric operator; Decomposition; Subdifferential; Semicontinuous convex function; Matrix; Fitzpatrick function; Fenchel conjugate; Linear relations
• Online Access: http://hdl.handle.net/2429/2807

We consider the decomposition of a maximal monotone operator into the sum of an antisymmetric operator and the subdifferential of a proper lower semicontinuous convex function. This is a variant of the well-known decomposition of a matrix into its symmetric and antisymmetric part. We analyze in detail the case when the graph of the operator is a linear subspace. Equivalent conditions of monotonicity are also provided. We obtain several new results on auto-conjugate representations including an explicit formula that is built upon the proximal average of the associated Fitzpatrick function and its Fenchel conjugate. These results are new and they both extend and complement recent work by Penot, Simons and Zălinescu. A nonlinear example shows the importance of the linearity assumption. Finally, we consider the problem of computing the Fitzpatrick function of the sum, generalizing a recent result by Bauschke, Borwein and Wang on matrices to linear relations.