Enveloping norms on the spaces of regularly P-operators in Banach lattices

We introduce and study the enveloping norms of regularly P-operators between Banach lattices E and F, where P is a subspace of the space L(E,F) of continuous operators from E to F. We prove that if P is closed in L(E,F) in the operator norm then the regularly P-operators forms a Banach space under the enveloping norm. Conditions providing that regularly P-operators forms a Banach lattice under the enveloping norm are given.