Solution of the quadratically hyponormal completion problem

For $m\ge 1$, let $\alpha : \alpha _{0}<\cdots <\alpha _{m}$ be a collection of ($m+1$) positive weights. The Quadratically Hyponormal Completion Problem seeks necessary and sufficient conditions on $\alpha$ to guarantee the existence of a quadratically hyponormal unilateral weighted shift $W$ with $\alpha$ as the initial segment of weights. We prove that $\alpha$ admits a quadratically hyponormal completion if and only if the self-adjoint $m\times m$matrix

is positive and invertible, where $q_{k}:=u_{k}+\vert s\vert^{2} v_{k}$, $r_{k}:=s\sqrt {w_{k}}$, $u_{k}:=\alpha _{k}^{2}-\alpha _{k-1}^{2}$, $v_{k}:=\alpha _{k}^{2}\alpha _{k+1}^{2}-\alpha _{k-1}^{2}\alpha _{k-2}^{2}$, $w_{k}:=\alpha _{k}^{2}(\alpha _{k+1}^{2}-\alpha _{k-1}^{2})^{2}$, and, for notational convenience, $\alpha _{-2}=\alpha _{-1}=0$. As a particular case, this result shows that a collection of four positive numbers $\alpha _{0}<\alpha _{1}<\alpha _{2}<\alpha _{3}$ always admits a quadratically hyponormal completion. This provides a new qualitative criterion to distinguish quadratic hyponormality from 2-hyponormality.