On the Maximal Monotone Operators in Hadamard Spaces

TL;DR

This paper extends monotone operator theory to Hadamard spaces by introducing p-Fenchel conjugates, examining the p-Fitzpatrick transform, and exploring relations with convex functions, advancing the understanding of these operators in non-linear metric spaces.

Contribution

It introduces the p-Fenchel conjugate and analyzes the p-Fitzpatrick transform for monotone operators in Hadamard spaces, providing new theoretical insights.

Findings

Established a Fenchel-Young inequality in Hadamard spaces

Analyzed properties of the p-Fitzpatrick transform

Linked maximal monotone operators with convex functions

Abstract

In this paper, some topics of monotone operator theory in the setting of Hadamard spaces are investigated. For a fixed element $p$ in a Hadamard space $X$, the notion of $p$-Fenchel conjugate is introduced and a type of the Fenchel-Young inequality is proved. Moreover, we examine the $p$-Fitzpatrick transform and its main properties for monotone set-valued operators in Hadamard spaces. Furthermore, some relations between maximal monotone operators and certain classes of proper, convex, l.s.c. extended real-valued functions on $X\times X^{\scalebox{0.7}{$^{\lozenge}$}}$, are given.