Second order cones for maximal monotone operators via representative functions (Q931417)

Let \(X\) be a reflexive Banach space and \(T:X \rightrightarrows X^{*}\) be a given maximal monotone operator. In this paper, the following formulas are proved: \[ \begin{aligned} T_M(z,z^*)^o &= \partial P''_T(z,z^*)(0,0)=\{(v,v^*)\mid F''_T(v,v^*)\leq 0\},\\ F''_T(v,v^*):&=\Bigl( b-e-\lim_{t\downarrow 0} 1/t^{2}\{F_T((z,z^{*})+ t(v,v^{*}))- F_{T}(z,z^{*})-t((v,v^{*}),(z,z^{*}))\}\Bigr)(v,v^{*})\\ &=F_{T_{M}(z,z^{*})}(v,v^{*}),\;\forall (v,v^{*})\in X\times X^{*}, \\ F''_{T}(v,v^{*}) &= F_{T_{M}(z,z^{*})}(v,v^{*}),\;\forall (v,v^{*})\in X\times X^{*},\end{aligned} \] where \[ \begin{aligned} T_{M}(z,z^{*}):&= b-\lim_{t\downarrow 0}1/t(M-(z,z^{*})),\\ F_{T}(z,z^{*})&= \sup_{(x,x^{*})\in \operatorname{Graph} T}\bigr\{\langle z,x^{*}\rangle+ \langle x,z^{*}\rangle-\langle x,x^{*}\rangle\bigr\},\\ T_{M}(z,z^{*})^{o}&= \bigl\{ (y^{*},y)\mid ((y,y^{*}),(h,h^{*}))\leq 0,\;\forall (h,h^{*})\in T_{M}(z,z^{*})\bigr\}. \end{aligned} \] Here, \(b-\lim_{t\downarrow 0}\) denotes convergence in the Attouch--Wets sense.