Reflexivity defect of spaces of linear operators

For a finite-dimensional linear subspace S⊆L(V,W) and a positive integer k, the k-reflexivity defect of S is defined by rdk(S)=dim(Refk(S)/S), where Refk(S) is the k-reflexive closure of S. We study this quantity for two-dimensional spaces of operators and for single generated algebras and their commutants.