# Hankel operator with symbol a Blaschke product

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).

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.
• By $B^*$ you mean $\bar{B}$? – Babyblog Oct 19 '17 at 10:07
• I have two more questions. If I consider $f(w)=w^n/B(w)$, where $B$ is a "common" Blaschke factor with $d$ poles $\beta_j \in \mathbb{D}$, can we have the explicit form of the optimal $h_{opti} \in H^{\infty}$ in Nehari's theorem: $\|H_f\| = \inf_{h\in H^{\infty}} \| f-h \|$? Is it true that if $f$ is rational then $h_{opti}$ is also rational? – Babyblog Oct 19 '17 at 12:34
• @Babyblog If $f$ is a rational function without zeros on the unit circle then the optimal $h$ exists and is also rational. There is an algorithm how to construct $h_{opt}$. Briefly, take the antianalytic part of $f$, do a state space realization, solve two Lyapunov equations and build $h_{opt}$ by Schmidt pair. The algorithm for continuous time is writen here, Ch, 5,6. – A.Γ. Oct 26 '17 at 8:30