Another example of an invariant subspace of \(H^{\infty}\) with index \(\mathfrak c\) (Q1409296)

Recall that a function \(\varphi\in H^\infty\) (\(H^\infty\) is the space of all bounded analytic functions on the open unit disc \(\mathbb{D}\)) satisfying \(|\varphi(e^{i\theta})|= 1\) almost everywhere on the unit circle \(\partial\mathbb{D}\) is said to be inner. Every inner function \(\varphi\) has the form \(\psi= e^{ic}b\psi\), where \(c\) is a real constant, \(b\) is a Blaschke product, and \(\psi\) is a singular inner function. \textit{A. Borichev} [J. Reine Angew. Math. 505, 23--44 (1998; Zbl 0923.47003)] gave an example of an invariant subspace \({\mathcal M}\) of \(H^\infty\) with \[ \dim{\mathcal M}/z{\mathcal M}= \text{card}[0, 1]= c, \] which is generated by an uncountable family of Balaschke products. In this survey, the author constructs singular inner functions which generate an invariant subspace \({\mathcal M}\) of \(H^\infty\) with \(\dim{\mathcal M}/z{\mathcal M}= \text{card}[0,1]= c\).