Entrywise Positivity Preservers on Green Matrices

We classify the entrywise functions that preserve positive semidefiniteness on discrete Green matrices $G(p,q)=(p_{\min (i,j)}{q}_{\max (i,j)})$ with positive parameters, without requiring the resulting matrix to retain Green structure. For matrices of all orders, the preservers are the zero function and the functions $f(t)={\int }_{[0,\mathrm{\infty })}{t}^{\alpha }d\mu (\alpha )$, where $\mu$ is a nonzero finite positive measure and the integral is finite for every $t>0$. Requiring the resulting matrix to be totally nonnegative reduces the nonzero preservers to $f(t)=c{t}^{\alpha }$, where $c>0$ and $\alpha \ge 0$. These power functions also preserve positive semidefinite Green structure, while strict Green structure is preserved precisely when $\alpha >0$. No regularity assumption is needed for these classifications. We also characterize continuously differentiable functions that are entrywise Loewner monotone on every fixed-$q$ Green family: this holds precisely when $f^{\mathrm{\prime }}$ is a positive mixture of nonnegative real powers, with the zero measure allowed.