Hankel operator with symbol a Blaschke product

Question

If $B={\prod}_j \varphi_j$ is a Blaschke product (finite or infinite) of Blaschke factors $\varphi_j(w)=\frac{w-\alpha_j}{1-\overline{\alpha_j}w}$ with $|\alpha_j|>1$, is it true that the norm of the Hankel operator (in Hardy spaces on the unit disk) $||H_B||$ is equal to one? I think I have proved it for a Blaschke factor but I do not see how to generalize it (if it is possible).

Answer 1

It is straightforward to see that $$ \|H_B\|=\|P_-B|H^2\|\le \|B\|_\infty=1. $$ Here $P_-$ is the projection onto $H_2^\bot$ On the other hand, $B^*h$ is analytic for every $h\in H^\infty$, and from the maximum modulus principle it follows that $$ \|1-B^*h\|_\infty\ge |1-\underbrace{B^*(z_i)}_{=0}h(z_i)|=1 $$ where $z_i$ is one of the zeros of $B^*$, for example, $1/\bar{\alpha_i}$. Then Nehari theorem implies that $$ \|H_B\|=\inf_{h\in H^\infty}\|B-h\|_\infty=\inf_{h\in H^\infty}\|1-B^*h\|_\infty\ge 1. $$ P.S. It is more common to call $B^*$ a Blaschke product - an analytic unimodular function with zeros inside the disc.