The fact that the Schur (that is, element wise) product of two positive definite (symmetric) matrices is positive definite immediately implies (using the convexity of the positive semi definite cone) that if $A$ is PSD, then so is $B=A\_f,$ such that $b\_{ij} = f(a\_{ij}),$ where $f$ is an analytic function all of whose coefficients are positive. The question is, is this result sharp, or are there other $f$ which maps the PSD cone into itself by being element wisely applied?



elementwise functions of positive definite matrix
If the matrices are real and the function you have in mind is real-valued, then you indeed get the characterization you suggested. This was first shown in "I. J. Schoenberg. Positive definite functions on spheres. Duke Math. J., 9:96-108, 1942" under the additional assumption that $f$ is continuous. As noted by UwF in an earlier comment, it was shown in [arxiv.org/abs/0709.1235](http://arxiv.org/abs/0709.1235) without the continuity assumption.


This can be generalized to matrices over $\mathbb{C}$ as well. You just have to be slightly careful, since there's another "obvious" element-wise map that preserves positive definiteness, but is not analytic: complex conjugation.


With that in mind, it is known that this is pretty much the only wrinkle that occurs in the complex case. It was shown in "C. S. Herz. Fonctions operant sur les fonctions definies-positives. Ann. Inst. Fourier (Grenoble), 13:161-180, 1963" that every element-wise function $f$ preserving positive definiteness is of the form


$$f(z) = \sum\_{i,j=0}^\infty c\_{ij} z^i \overline{z^j}$$


with this series converging everywhere and each $c\_{ij} \geq 0$.


