Let $G$ be a connected domain in $\mathbb{C}^{n}$, let $H$ be a Hilbert space and let $f,g:G\to H\backslash \{0\}$ be holomorphic (in my particular situation they are also injective, but I don't think it helps).

Is it true that if $\frac{\partial^2}{\partial z_i\partial \overline{z_j}}\log \|f(z)\|=\frac{\partial^2}{\partial z_i\partial \overline{z_j}}\log \|g(z)\|$, for all $i,j\le n$, then there is a holomorphic function $h:G\to\mathbb{C}$ and an isometry $U:H\to H$, such that $g(z)=h(z)Uf(z)$?

Motivation. Consider the canonical quotient $p:H\to PH$, where the latter is the projective space over $H$. Suppose $\varphi:G\to G$ is a biholomorphism, such that $f$ and $g=f\circ\varphi$ satisfy that condition. Then $pf\varphi(pf)^{-1}$ is an isometry of $pf(G)$ with respect to the Fubini-Study metric on $PH$. I wonder, if this isometry can be lifted to an isometry $H$, i.e. if there is an isometry $U:H\to H$, such that $pUf=pf\varphi$. Finally the last condition means that there is a holomorphic function $h:G\to\mathbb{C}$, such that $h(z)Uf(z)=f(\varphi(z))$.