A necessary condition for certain functions to preserve positive semi-definiteness on partitioned matrices

If $f$ is a symmetric complex-valued function on the $m$-fold Cartesian product of the set of non-negative reals and $A$ is a positive semi-definite $m\times m$ matrix with eigenvalues $λ_j$, we set $f(A):=f(λ_1,\dotsc,λ_m)$. It is shown that if $[f(A_{αβ})]$ is positive semi-definite whenever $[A_{αβ}]$ is a positive semi-definite matrix with positive semi-definite entries $A_{αβ}$, then $f$ has a power series expansion with positive coefficients.

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