Monotone Relations in Hadamard Spaces

Abstract

In this paper,  the notion of $\mathcal{W}$-property for subsets of $X\times X^\loze$ is introduced  and investigated, where $X$ is an Hadamard space and $X^\loze$ is its linear dual space. It is shown that an Hadamard space $X$ is flat if and only if $X\times X^\loze$ 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.