The starting place is a brief proof of a well-known result, the hyponormality of C_k (the generalized Cesàro operator of order one) for k ≧ 1. This leads to the definition of a superclass of the posinormal operators. It is shown that all the injective unilateral weighted shifts belong to this superclass.

Sufficient conditions are determined for an operator in this superclass to be posinormal and hyponormal. A connection is established between this superclass and some recently-published sufficient conditions for a lower triangular factorable matrix to be a hyponormal bounded linear operator on ℓ².