Tensor products and property (w)

A Banach space operator satisfies “property (w)” if the complement of its essential Weyl approximate point spectrum in its approximate point spectrum is the set of finite multiplicity isolated eigenvalues of the operator. Property (w) does not transfer from operators A and B to their tensor product A?B; we give necessary and/or sufficient conditions ensuring the passage of property (w) from A and B to A?B. Perturbations by Riesz operators are considered.