Weighted holomorphic spaces with trivial closed range multiplication operators

We deal with the space $H^{\infty}_{v}$ consisting of those analytic functions $f$ on the unit disc $\mathbb{D}$ such that $\Vert f\Vert _v := \sup_{z \in \mathbb{D} } v(z) \vert f(z) \vert<\infty$, with $v(z)=v(\vert z\vert)$. We determine the critical rate of decay of $v$ such that the pointwise multiplication operator $M_{\varphi}$, $M_{\varphi}(f)(z)=\varphi(z) f(z)$ and $\varphi$ analytic, has closed range in $H^{\infty}_{v}$ only in the trivial case that $\varphi$ is the product of an invertible function in $H^\infty$ and a finite Blaschke product.