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

TL;DR

This paper proves that a specific weighted mean matrix with linear positive weights is posinormal and coposinormal on ℓ², leading to its hyponormality and suggesting broader linear cases.

Contribution

It establishes the posinormality and coposinormality of weighted mean matrices with linear positive weights, including a specific case with odd integers, and proposes a conjecture for general linear weights.

Findings

Weighted mean matrix with weights $w_n=2n+1$ is hyponormal.

The operator is shown to be posinormal and coposinormal.

The results suggest broader applicability to linear weight sequences.

Abstract

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.