Characterization of the restricted type spaces R(X)

We study functorial properties of the spaces $R(X)$, introduced in [Studia Math. 197 (2010), 69-79] as a central tool in the analysis of the Hardy operator minus the identity on decreasing functions. In particular, we provide conditions on a minimal Lorentz space ${\mathrm{\Lambda }}_{\phi }$ so that the equation $R(X)={\mathrm{\Lambda }}_{\phi }$ has a solution within the category of rearrangement invariant (r.i.) spaces. Moreover, we show that if $R(X)={\mathrm{\Lambda }}_{\phi }$, then we can always take $X$ to be the minimal r.i. Banach range space for the Hardy operator defined in ${\mathrm{\Lambda }}_{\phi }$.