Type: Article
Publication Date: 2021-01-01
Citations: 20
DOI: https://doi.org/10.1353/ajm.2021.0049
Given a domain $I\subset\Bbb{C}$ and an integer $N>0$, a function $f: I\to\Bbb{C}$ is said to be {\it entrywise positivity preserving} on positive semidefinite $N\times N$ matrices $A=(a_{jk})\in I^{N\times N}$, if the entrywise application $f[A]=(f(a_{jk}))$ of $f$ to $A$ is positive semidefinite for all such $A$. Such preservers in all dimensions have been classified by Schoenberg and Rudin as being absolutely monotonic. In fixed dimension $N$, results akin to work of Horn and Loewner show that the first $N$ non-zero Maclaurin coefficients of any positivity preserver $f$ are positive; and the last $N$ coefficients are also positive if $I$ is unbounded. However, very little was known about the higher-order coefficients: the only examples to date for unbounded domains $I$ were absolutely monotonic, hence work in all dimensions; and for bounded $I$ examples of non-absolutely monotonic preservers were very few (and recent).