A Note on Joint Hyponormality

We describe certain cones of polynomials in two variables naturally associated to the class(es) of operators $T$ for which the tuple $(T,{T^2}, \ldots ,{T^n})$ is jointly (weakly) hyponormal. As an application we give an example of an operator $T$ such that the tuple $(T,{T^2})$ is jointly but not weakly hyponormal. Further, we show that there exists a polynomially hyponormal operator which is not subnormal if and only if there exists a weighted shift with the same property.