In this paper we study the
question: which compact operators are contained in ℛ(δS)−, the norm closure of the
range of the derivation δS(X) = SX − XS induced by a weighted shift S? We find
that ℛ(δS)− always contains the lower triangular (with respect to the basis (ei) on
which S is a shift) compact operators. Further, ℛ(δS)− contains the n-lower
triangular (operators T satisfying (Tei,ej) = 0 for i − j > n) compact operators
if and only if e1⊗ en+1∈ℛ(δS)−. We also find necessary and sufficient
conditions on the weights of S in order that e1⊗ en+1∈ℛ(δS)− and that 𝒦,
the algebra of compact operators, be contained in ℛ(δS)−. These results
completely answer the question: which essentially normal weighted shifts are
d-symmetric?
