On Fitzpatrick functions of monotone linear operators

Abstract

Bauschke, Borwein and Wang have shown in [H.H. Bauschke, J.M. Borwein, X. Wang, Fitzpatrick functions and continuous linear monotone operators, Siam J. Optimization, 18 (3) (2007), 789–809] that if FT(⋅,⋅) denotes the Fitzpatrick function of a continuous linear monotone operator T on a separable real Hilbert space H, then FT(x,u)=2qT+∗((u+T∗x)/2), where qA∗ denotes the Fenchel conjugate of the function qA:H→R sending x to 2−1〈x,Ax〉 for an arbitrary continuous positive symmetric operator A∈B(H). Here, T+≔(T+T∗)/2≥0 and A† denotes the Moore–Penrose type inverse of a positive symmetric operator A. The main result of the present paper is a sharpening of the result, achieved by showing that dom(qA∗)=ran(A1/2) and that FT(x,u)=4−1‖(T+1/2)†(u+T∗x)‖2 on dom(qA∗), where dom(f) is the set on which an extended real-valued function f is finite. We will also find a formula for the Fitzpatrick function of a general n-cyclic monotone linear operator in terms of a corresponding ordinary monotone linear operator.