Unique direct-sum decompositions of infinite-dimensional unitary representations of the infinite dihedral group via real CMV matrices

Unitary representations of the infinite dihedral group arise naturally and play an important role in a wide range of areas including mathematics, physics, and engineering. CMV matrices, introduced by Cantero, Moral, and Velázquez, are five-diagonal unitary matrices that are standard tools in the theory of orthogonal polynomials on the unit circle. It is known that every unitary representation of the infinite dihedral group on a separable Hilbert space admits a discrete direct-sum decomposition into representations arising from the natural factorizations of real CMV matrices and copies of two exceptional one-dimensional irreducible representations. We prove that such a decomposition is unique up to unitary equivalence when the summands arising from real CMV matrices are grouped according to their multiplicities and the spectral measures of the corresponding matrices are mutually singular across distinct multiplicities. Furthermore, we show that the representation associated with each such real CMV matrix $C$ admits a direct integral decomposition into irreducible unitary representations with respect to the push-forward of the restriction of the spectral measure of $C$ to the closed upper half circle. These results make it possible to study the structure of unitary representations of the infinite dihedral group using CMV matrices and the theory of orthogonal polynomials on the unit circle.