On Non-Enlargeable and Fully Enlargeable Monotone Operators

We consider a family of enlargements of maximal monotone operators in a reflexive Banach space. Each enlargement, depending on a parameter

, is a continuous point-to-set mapping

E(\varepsilon,x) E ( ε , x )

whose graph contains the graph of the given operator

T T

. The enlargements are also continuous in

\varepsilon ε

, and they coincide with

T T

for

\varepsilon=0 ε = 0

. The family contains both a maximal and a minimal enlargement, denoted as

T^e T e

and

T^{se} T s e

respectively. We address the following questions: a) which are the operators which are not enlarged by

T^e T e

, i.e., such that

T(\cdot)=T^e(\varepsilon,\cdot) T ( ⋅ ) = T e ( ε , ⋅ )

for some

\varepsilon>0 ε > 0

? b) same as (a) but for

T^{se} T s e

instead of

T^e T e

. c) Which operators are fully enlargeable by

T^e T e

, in the sense that for all

x x

and all

\varepsilon>0 ε > 0

there exists

\delta>0 δ > 0

such that all points whose distance to

T(x) T ( x )

is less than

\delta δ

belong to

T^e(\varepsilon,x) T e ( ε , x )

? We prove that the operators not enlarged by

T^e T e

are precisely the point-to-point affine operators with skew symmetric linear part; those not enlarged by

T^{se} T s e

are the point-to-point and affine operators, and the operators fully enlarged by

T^e T e

are those operators

T T

whose Fitzpatrick function is continuous in its second argument at pairs belonging to the graph of

T T

.