A corona theorem for countably many functions

AbstractSuppose a = {aj}1∞ is a sequence of H∞ functions on the unit disk D such that $$||a||_\infty = \mathop {\sup }\limits_{z \in D} (\sum\limits_1^\infty { |a_j (z)|^2 } )^{{\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}}< \infty $$ . We show that there exists a sequence c = {cj}∞1 of H∞ functions with ∥c∥∞ < ∞ and satisfying $$\sum\limits_1^\infty {a_j c_j } = 1$$ on D if and only if $$\mathop {\inf }\limits_{z \in D} \sum\limits_1^\infty { |a_j (z)|^2 } > 0$$ .