Weak infinite products of Blaschke products (Q2750857)

Let \(B= (b_1,b_2,\dots)\) be a sequence of infinite Blaschke products and \(\{z_{k,j}\}\) be the zeros of \(b_k\). \({\mathcal N}(B)\) denotes the set of sequences \(N= (n_1,n_2,\dots)\) such that \(\sum^\infty_{k=1} \sum^\infty_{j= n_k}(1-|z_{k,j}|)< \infty\). Put \(B_N= \prod^\infty_{k=1} \prod^\infty_{j= n_k} {-\overline z_{k,j}\over|z_{k,j}|} {z- z_{k,j}\over 1-\overline z_{k,j}z}\) and \(N\in{\mathcal N}(B)\). \(B_N\) is called a weak infinite product of Blaschke products \(\{b_k\}_k\). Let \(H^\infty\) be the usual Hardy space on the unit circle \(\partial D\) and \(C\) the space of all continuous functions on \(\partial D\). \(M(H^\infty+ C)\) denotes the maximal ideal space of \(H^\infty+C\) and \(Z(f)= \{\zeta\in M(H^\infty+ C); f(\zeta)= 0\}\). The author shows the following interesting result: Let \(f\in H^\infty+C\), \(f\neq 0\) and \(\text{int }Z(f)\neq \varnothing\). Let \(E\) be on \(F_\sigma\)-subset of \(M(H^\infty+ C)\) such that \(E\subset \text{int }Z(f)\). Then \(\overline E\subset \text{int }Z(f)\). Using this result, he solves an open problem posed by \textit{P. Gorkin} and \textit{R. Mortini} [Indiana Univ. Math. J. 49, No. 1, 287-309 (2000; Zbl 0968.46036)].