The weighted mean matrix with weight sequence $w_n=2n+1$ is a hyponormal   operator on $\ell^2$
  A weighted mean matrix whose weight sequence is linear with positive coefficients is shown to be a posinormal operator on $\ell^2$. This operator is also shown to be coposinormal, so it and its adjoint have the same null space and the same range. The posinormality result leads to a proof that the weighted mean matrix associated with the sequence of odd positive integers is hyponormal, as well as a conjecture regarding a more general linear case. 
H. C. Rhaly Jr
math.FA