An unbounded product \(T=AB\) of two nonnegative selfadjoint operators A and B, where \(\rho (T) \ne \emptyset \) and A is bounded, is proved to have the single valued extension property (SVEP) and, more significantly, to be a \(\mathbb {C}\) -generalized scalar operator. This property plays a central role as, it not only implies the Dunford’s property for the product AB, but it also allows any operator T that is quasi-affine to a nonnegative selfadjoint operator S to satisfy \(\sigma (T)=\sigma (S)\) . These results have required a deeper study of the spectral connection between operators S having SVEP and Dunford properties and operators T that are quasi-affine to S.
