  The starting place is a brief proof of a well-known result, the hyponormality
of $C_k$ (the generalized Ces\`{a}ro operator of order one) for $k \geq 1$.
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 $\ell^2$.
