Multiplication operators on weighted Banach spaces of analytic functions with exponential weights (Q2777737)

Let \(v\) be a radial weight on the unit disc \(\mathbb{D}\) such that for some \(\varepsilon>0\), \(-(1-|z|^2)^{\varepsilon+2} \Delta\log v(z) \to+ \infty\) as \(z\) tends to the boundary \((\Delta\) is the Laplacian in \(\mathbb{R}^2)\). This survey gives some necessary and sufficient conditions for a multiplication operator \(M_\varphi\) to be an isomorphism on the following weighted space: NEWLINE\[NEWLINEH_v^\infty= \biggl\{f\in H(\mathbb{D}):\|f\|_v= \sup_{z\in \mathbb{D}}v(z) \bigl|f(z)\bigr|<\infty\biggr\}.NEWLINE\]NEWLINE These necessary and sufficient conditions can be, in fact, expressed by \(\varphi=hb\) where \(h\in H^\infty (\mathbb{D})\) and \(b\) is a Blaschke product. Moreover, for the weight \(v\), the approximate point spectrum of \(M_\phi\) is identified as \(\varphi (M(H^\infty) \setminus\mathbb{D})\), where \(M(H^\infty)\) is the maximal ideal space of \(H^\infty\).