Enveloping norms of regularly P-operators in Banach lattices

The span of positive linear operators belonging to an operator linear class P and acting between Banach lattices is rarely a Banach space under the operator norm. We investigate the enveloping norm \(\Vert S\Vert _{\text {r-P}}=\inf \{\Vert T\Vert : \pm S\le T\in \text {P}\}\) ${\Vert S\Vert }_{\text{r-P}}=inf\{\Vert T\Vert :\pm S\le T\in \text{P}\}$ on \({\text {span}}(\text {P}_+(E,F))\) $\text{span}({\text{P}}_{+}(E,F))$ that is complete under rather mild assumptions on P.