Open AccessFitzpatrick Functions and Continuous Linear Monotone Operators
Author(s) -
Heinz H. Bauschke,
Jonathan M. Borwein,
Xianfu Wang
Publication year - 2007
Publication title -
siam journal on optimization
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.066
H-Index - 136
eISSN - 1095-7189
pISSN - 1052-6234
DOI - 10.1137/060655468
Subject(s) - mathematics , monotonic function , monotone polygon , hilbert space , strongly monotone , operator (biology) , subderivative , pseudo monotone operator , pure mathematics , operator theory , euclidean space , operator norm , discrete mathematics , regular polygon , convex optimization , finite rank operator , mathematical analysis , banach space , operator space , biochemistry , chemistry , geometry , repressor , transcription factor , gene
The notion of a maximal monotone operator is crucial in optimization as it captures both the subdifferential operator of a convex, lower semicontinuous, and proper function and any (not necessarily symmetric) continuous linear positive operator. It was recently discovered that most fundamental results on maximal monotone operators allow simpler proofs utilizing Fitzpatrick functions. In this paper, we study Fitzpatrick functions of continuous linear monotone operators defined on a Hilbert space. A novel characterization of skew operators is presented. A result by Brézis and Haraux is reproved using the Fitzpatrick function. We investigate the Fitzpatrick function of the sum of two operators, and we show that a known upper bound is actually exact in finite-dimensional and more general settings. Cyclic monotonicity properties are also analyzed, and closed forms of the Fitzpatrick functions of all orders are provided for all rotators in the Euclidean plane
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