Interpolating Blaschke products and ideals of the algebra $H^\infty$

V. A. Tolokonnikov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XII, Tome 126 (1983), pp. 196-201

Résumé

For a function $f$ in $H^\infty(l^2)$ the ideals $I(f)=\{h\in H^\infty:h=\sum_{i=1}^\infty f_ig_i, g\in H^\infty(l^2)\}$ and $J(f)=\{h\in H^\infty:|h(z)|\leqslant c\|f(z)\|_2, z\in\mathbb D\}$ are considered. The functions $f$ for which there exists an interpolating Blaschke product in $I(f)$ (or $J(f)$) are characterized. Moreover there is given a characterization of functions $u$ in $H^\infty$ for which $$ f\in H^\infty(l^2), u\in J(f)\Rightarrow u\in I(f). $$ (In the case $u=1$ the latter implication is the Carleson Corona theorem).