Given a sequence \(\{a_k\}_{k\in\mathbb Z}\subset\mathbb D\) one can construct the matrices \[ \Theta_k=\begin{bmatrix} \overline{a_k}&\sqrt{1-|a_k|^2}\\ \sqrt{1-|a_k|^2}&-a_k\end{bmatrix}. \] These can be seen as operators on \(\operatorname{span}\{\delta_k,\delta_{k+1}\}\subset\ell^2(\mathbb Z)\). Define \(L,M,C\in B(\ell^2(\mathbb Z))\) by \[ L=\bigoplus_{k\in\mathbb Z}\Theta_{2k},\qquad M=\bigoplus_{k\in\mathbb Z}\Theta_{2k+1},\qquad C=LM. \] The set \(\mathcal T_{\mathrm{CMV}}(E)\) is defined as consisting of those \(C\) that have spectrum made of a fixed finite union of disjoint non-degenerate closed circular arcs \(E\subset\partial\mathbb D\), and such that \(\{a_k\}\) is almost periodic.
By means of a Möbius transform, the authors define the class of MCMV matrices. They depend further on parameters \(\vec z\in\mathbb D^n\) and \(\vartheta\in\mathbb R/2\pi\mathbb Z\). The class \(\mathcal T_{\mathrm{MCMV}}(E,\vec z,\lambda_*)\), where \(\lambda_*\in\partial\mathbb D\setminus E\), is defined as the set of those MCMV matrices with spectrum \(E\) and discriminant \(\Delta_A(\lambda_*)\). The discriminant \(\Delta_A(z)\) is defined similarly to the CMV case in [D. Damanik et al., Ann. Math. (2) 171, No. 3, 1931–2010 (2010; Zbl 1194.47031)], and the analog “magic formula” is stated and proved by the authors: \[ A\in\mathcal T_{\mathrm{MCMV}}(E,\vec z,\lambda_*)\iff \Delta_E(A)=S^{2(g+1)}+S^{-2(g+1)}, \] where \(S\) is the bilateral shift and \(g+1\) is the number of arcs in \(E\). The authors also show that \[ \mathcal T_{\mathrm{CMV}}(E)\simeq\mathcal T_{\mathrm{MCMV}}(E,\vec z,\lambda_*) \] via a unitary bijection.
A reformulation of the above results allows the authors to solve a conjecture of B. Simon [Orthogonal polynomials on the unit circle. Part 2: Spectral theory. Providence, RI: American Mathematical Society (2005; Zbl 1082.42021)]. Namely, they show that for any \(C\in\mathcal T_{\mathrm{CMV}}(E)\) the Carathéodory function \(F_+\) associated to the half-line restriction \(C_+\) is a quadratic irrationality, which means that there exist polynomials \(a(z),b(z),c(z)\) such that \(F_+\) solves \[ a(z)F_+(z)^2+b(Z)F_+(z)+c(z)=0. \] The proofs of the above results are obtained after developing significant partial results that cover 44 pages of the paper.