Maximal monotone operators with non-maximal graphical limit

TL;DR

This paper demonstrates that the graphical limit of maximally monotone operators may not be maximally monotone and explores the conditions under which the resolvent's proto-derivative is maximally monotone, advancing understanding of operator limits.

Contribution

It provides a counterexample for the non-maximality of graphical limits and characterizes the resolvent's differentiability via proto-derivatives, offering new insights into monotone operator theory.

Findings

Counterexample showing non-maximality of graphical limits

Characterization of resolvent differentiability via proto-derivatives

Conditions linking maximal monotonicity of proto-derivatives to operator limits

Abstract

We present a counterexample showing that the graphical limit of maximally monotone operators might not be maximally monotone. We also characterize the directional differentiability of the resolvent of an operator $B$ in terms of existence and maximal monotonicity of the proto-derivative of $B$.