Algebras on the disk and doubly commuting multiplication operators

Abstract:

We prove that a bounded analytic function $f$ on the unit disk is in the little Bloch space if and only if the uniformly closed algebra on the disk generated by ${H^\infty }$ and $\overline f$ does not contain the complex conjugate of any interpolating Blaschke product. A version of this result is then used to prove that if $f$ and $g$ are bounded analytic functions on the unit disk such that the commutator ${T_f}T_g^{\ast } - T_g^{\ast }{T_f}$ (here ${T_f}$ denotes the operator of multiplication by $f$ on the Bergman space of the disk) is compact, then $(1 - |z{|^2})\min \{ |f’ (z)|,\;|g’ (z)|\} \to 0$ as $|z| \uparrow 1$.