Darboux transformations for CMV matrices

TL;DR

This paper develops a theory of Darboux transformations for CMV matrices, revealing their properties, differences from Jacobi matrices, and connections to orthogonal polynomials and measure modifications.

Contribution

It introduces Darboux transformations for CMV matrices, explores their properties, and uncovers unique features and challenges compared to Jacobi matrices.

Findings

Darboux transformations for CMV matrices are linked to Laurent polynomial modifications of measures.

Inverse Darboux transformations can produce spurious solutions that are not unitary or band matrices.

The theory is based on Cholesky factorizations of Hermitian Laurent polynomials evaluated on CMV matrices.

Abstract

We develop a theory of Darboux transformations for CMV matrices, canonical representations of the unitary operators. In perfect analogy with their self-adjoint version -- the Darboux transformations of Jacobi matrices -- they are equivalent to Laurent polynomial modifications of the underlying measures. We address other questions which emphasize the similarities between Darboux transformations for Jacobi and CMV matrices, like their (almost) isospectrality or the relation that they establish between the corresponding orthogonal polynomials, showing also that both transformations are connected by the Szeg\"o mapping. Nevertheless, we uncover some features of the Darboux transformations for CMV matrices which are in striking contrast with those of the Jacobi case. In particular, when applied to CMV matrices, the matrix realization of the inverse Darboux transformations -- what we call 'Darboux transformations with parameters' -- leads to spurious solutions whose interpretation deserves future research. Such spurious solutions are neither unitary nor band matrices, so Darboux transformations for CMV matrices are much more subject to the subtleties of the algebra of infinite matrices than their Jacobi counterparts. A key role in our theory is played by the Cholesky factorizations of infinite matrices. Actually, the Darboux transformations introduced in this paper are based on the Cholesky factorizations of degree one Hermitian Laurent polynomials evaluated on CMV matrices. We also show how these transformations can be generalized to higher degree Laurent polynomials, as well as to the extension of CMV matrices to quasi-definite functionals -- what we call 'quasi-CMV' matrices.