Preprint A374/2005
On non-enlargeable and fully enlargeable monotone operators

Alfredo N. Iusem | Burachik, Regina S.

**Keywords: **
maximal monotone operators | enlargements

In this paper we consider a family of enlargements of maximal monotone operators in a reflexive Banach space. Each enlargement, depending on a non-negative real parameter e, is a continuous point-to-set mapping E(e,x) whose graph contains the graph of the given operator T. The enlargments are also continuous in e, and they coincide with T for e = 0. The family contains a maximal and a minimal element, denoted T^M and T^m respectively. We address the following questions:
a) which are the operators which are not enlarged by T, i.e.
such that T(.) = T^M(e,.) for some e > 0?
b) same as (a), but for T^m instead of T^M.
c) Which operators are fully enlargeable by T^M, in the sense
that for all x and for all e > 0 there exists d > 0 such
that all points whose distance to T(x) is less than d belong
to T^M(e,x)?
We prove that the operators not enlarged by T^M are precisely the point-to-point affine operators with skew-symmetric linear part; those not enlarged by T^m are the point-to-point and affine operators, and the operators fully enlarged by T^M ate those operators T whose Ftizpatrick function is continuous in its second argument at pairs belonging to the graph of T.