On the stability index for weighted composition operators

Let ϵ>0ϵ>0. A continuous linear operator T:C(X)⟶C(Y)T:C(X)⟶C(Y) is said to ϵϵ-preserve disjointness   if ‖(Tf)(Tg)‖∞≤ϵ‖(Tf)(Tg)‖∞≤ϵ, whenever f,g∈C(X)f,g∈C(X) satisfy ‖f‖∞=‖g‖∞=1‖f‖∞=‖g‖∞=1 and fg≡0fg≡0. In this paper we continue our study of the minimal interval where the possible maximal distance from a norm one operator which ϵϵ-preserves disjointness to the set of weighted composition maps may lie. We provide sharp bounds for both the finite and the infinite case, which turn out to be completely different.