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A weighted mean matrix whose weight sequence is linear with positivecoefficients is shown to be a posinormal operator on $\ell^2$. This operator isalso shown to be coposinormal, so it and its adjoint have the same null spaceand the same range. The posinormality result leads to a proof that the weightedmean matrix associated with the sequence of odd positive integers ishyponormal, as well as a conjecture regarding a more general linear case.

The weighted mean matrix with weight sequence $w_n=2n+1$ is a hyponormal operator on $\ell^2$