The complement of M(a) in H(b)

Abstract

Let $b$ be a nonextreme function in the unit ball of $H^{\infty}$ on the unit disk $\mathbb{D} $ and let $a$ be an outer $H^{\infty}$ function such that $|a|^2+|b|^2=1$ almost everywhere on $\partial \mathbb{D}$. The sufficient and necessary conditions for the orthogonal complement of $\mathcal{M}(a)$ in $\mathcal{H}(b)$ be finite dimensional has been given by D. Sarason . Here we describe this space explicitly.