Let $G$ be a connected simply connected domain in $\mathbb{C}^{n}$, let $H$ be a Hilbert space.

Q1. Which functions $F:G\to(0,+\infty)$ are such that there is a holomorphic $f:G\to H\backslash \{0\}$, such that $F(z)=\|f(z)\|$?

I guess that plurisubharmonicity is necessary, but is it sufficient?

Note that $f$ is "almost" uniquely determined by $F$ in the sense that if $\|f(z)\|=\|g(z)\|$, for every $z\in G$, for another holomorphic $g:G\to H\backslash \{0\}$, then there is an isometry $U:H\to H$, such that $g(z)=Uf(z)$ (see also my previous question). In particular $K(x,y)=\left<f(x),f(y)\right>$ is completely determined by $F$. This leads to the second question.

Q2. Is there a formula that expresses $K$ in terms of $F$?