Monotone Relations in Hadamard Spaces

Abstract

In this paper, the notion of $\mathcal{W}$-property for subsets of $X\times X^{◊}$ is introduced and investigated, where $X$ is an Hadamard space and $X^{◊}$ is its linear dual space. It is shown that an Hadamard space $X$ is flat if and only if $X\times X^{◊}$ has $\mathcal{W}$-property. Moreover, the notion of monotone relation from an Hadamard space to its linear dual space is introduced. Finally, a characterization result for monotone relations with $\mathcal{W}$-property (and hence in flat Hadamard spaces) is proved.