On Fitzpatrick functions of monotone linear operators

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 F T ( ⋅ , ⋅ ) denotes the Fitzpatrick function of a continuous linear monotone operator T on a separable real Hilbert space H , then F T ( x , u ) = 2 q T + ∗ ( ( u + T ∗ x ) / 2 ) , where q A ∗ denotes the Fenchel conjugate of the function q A : H → R sending x to 2 − 1 〈 x , A x 〉 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 ( q A ∗ ) = ran ( A 1 / 2 ) and that F T ( x , u ) = 4 − 1 ‖ ( T + 1 / 2 ) † ( u + T ∗ x ) ‖ 2 on dom ( q A ∗ ) , 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.