Say that a $\omega\times \omega$ Hermitian matrix $A$ is positive semidefinite of rank $n$ if there exists a $\omega\times n$ complex matrix $B$ such that $A=B B^\dagger$ where $^\dagger$ denotes the conjugate transpose.

Let $f$ be a real-analytic function that converges in a neighbourhood of the origin in ${\mathbb{C}}$. Develop $f=\sum_{i,j=0}^\infty c_{ij} z^i\bar{z}^j$ as a power series in $z$ and $\bar{z}$. Suppose that $f$ is real-valued so that $(c_{ij})$ is a $\omega\times \omega$ Hermitian matrix.

Suppose one shows that for any $(a_k)\in l^2({\mathbb{C}})$, the sum $\sum_{i,j=0}^\infty c_{ij} a_i\bar{a_j}$ is nonnegative. Does this imply that $(c_{ij})$ is positive semidefinite of some rank $n\le \omega$?

This characterization of positive semidefiniteness is valid for finite rank Hermitian matrix. But I'm unsure about the convergence conditions in the infinite rank case.