A corona theorem for countably many functions

Abstract

Suppose a = {a_j} ^∞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 = {c_j} ¹_∞ of H^∞ functions with ∥c∥_∞ < ∞ and satisfying$\sum\limits_1^\infty {a_j c_j } = 1$ on D if and only if$\mathop {\inf }\limits_{z \in D} \sum\limits_1^\infty { |a_j (z)|^2 } > 0$.