Weighted holomorphic spaces with trivial closed range multiplication operators

Abstract:

We deal with the space $H_v^\infty$ consisting of those analytic functions $f$ on the unit disc $\mathbb {D}$ such that $\|f\|_v := \sup _{z \in \mathbb {D} } v(z) |f(z) |<\infty$, with $v(z)=v(|z|)$. 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.