Historically, the notion of Group Equivariant Nonexpansive Operators (GENEOs) has been studied in Banach spaces \((\Phi (X),\Vert \cdot \Vert )\) of bounded real-valued functions defined on a set X. Our goal is to explore these objects in pseudometric spaces \((\Phi (X),d_{H})\) of real-valued functions, where \(d_{H}\) denotes the projective Hilbert metric on the cone \(\Phi ^{+}(X)\) of positive functions and appropriately extended to all \(\Phi (X)\) . It is important to note that, in our framework, the class \(\Phi (X)\) is admissible with respect to the pseudometric \(d_{H}\) , that is, \(d_{H}(\phi \circ g,\chi \circ g)=d_{H}(\phi ,\chi )\) for all \(\phi ,\chi \in \Phi (X)\) and \(g\in G\) , where G denotes a subgroup of the group B(X) of all bijections of X, provided that \(\phi \circ g\in \Phi (X)\) for all \(\phi \in \Phi (X)\) and \(g\in G\) . We show that the GENEOs can be seen as homogeneous order preserving mappings. Furthermore, we provide a significant example of a linear GENEO on \(\Phi ^{+}(X)\) : the weighted Banach operator (WBO) in which the weight satisfies certain invariance properties. These WBOs can be extended to the entire pseudometric space \((\Phi (X),d_{H})\) , while preserving the GENEO property, by means of an operator extension procedure introduced in the article. In this particular case, the procedure can be implemented by taking advantage of the properties of composition operators whose symbols are bijections. For instance, this extension procedure is always applicable to admissible classes \(\Phi (X)\) when X is a finite set, which is an important situation in applications. The article provides several examples to clarify and support the main notions and results. Finally, we also discuss the relationship between pseudometric GENEOs and averaged operators.
