The corona theorem for multiplier algebras on weighted Dirichlet spaces (Q1951355)

For \(\alpha\in \;]0,1]\), let \(D_\alpha\) be the weighted Dirichlet space given by \[ \bigg\{f(z)=\sum_{n=0}^\infty a_n z^n \;\Bigl| \Bigr.\;\; ||f||^2_\alpha:=\sum_{n=0}^\infty (n+1)^\alpha|a_n|^2<\infty\bigg\}. \] The associated multiplier algebra \(\mathcal M(D_\alpha)\) is the set of all functions \(\phi\in D_\alpha\) such that \(\phi f\in D_\alpha\) for all \(f\in D_\alpha\). The authors prove the following corona-type theorem: There is a constant \(C_\alpha<\infty\) so that for any sequence \(F:=(f_j)\) in \(\mathcal M(D_\alpha)\) with \[ \| M^C_F\|\leq 1\quad\text{and}\quad 0<\epsilon^2 \leq \sum_{j=1}^\infty |f_j(z)|^2\quad\text{for all}\quad z\in \mathbb D, \] there exists a sequence \(G:=(g_j)\) in \(\mathcal M(D_\alpha)\) such that \[ \sum_{j=1}^\infty f_jg_j=1\quad\text{and}\quad \| M_G^C\|\leq C_\alpha \epsilon^{-4}, \] where \[ M^C_F: D_\alpha\to \bigoplus_{n=1}^\infty D_\alpha, \quad M^C_F(h)=(f_1h,f_2h,\dots)^\perp, \] is the associated column operator. The proof uses Hilbert-space methods (as the commutant lifting theorem.)