The local equicontinuity of a maximal monotone operator

TL;DR

This paper investigates the conditions under which a maximal monotone operator in a barreled locally convex space exhibits local equicontinuity, extending previous results related to Fitzpatrick functions and algebraic interior points.

Contribution

It provides direct consequences of existing theorems on local equicontinuity for maximal monotone operators in barreled locally convex spaces.

Findings

Local equicontinuity holds on the algebraic interior of the projection of the Fitzpatrick function's domain.

The paper derives new implications for maximal monotone operators based on prior theoretical results.

It clarifies the relationship between Fitzpatrick functions and local equicontinuity in the context of barreled spaces.

Abstract

The local equicontinuity of an operator $T:X\rightrightarrows X^{*}$ with proper Fitzpatrick function $\varphi_{T}$ and defined in a barreled locally convex space $X$ has been shown to hold on the algebraic interior of $\operatorname*{Pr}\,_{X}(\operatorname*{dom}\varphi_{T})$). The current note presents direct consequences of the aforementioned result with regard to the local equicontinuity of a maximal monotone operator defined in a barreled locally convex space.