Friday, January 4, 2013

1301.0501 (Paul Munger et al.)

An expression for the Green's function of the two-sided CMV matrix    [PDF]

Paul Munger, Darren C. Ong
The doubly infinite CMV matrix is a unitary operator acting on $\ell^2(\mathbb Z)$ that is important in the study of orthogonal polynomials on the unit circle and in quantum random walks. We exhibit an expression for the resolvent of the CMV matrix in terms of the resolvents of the two semi-infinite CMV matrices that comprise its two "halves". This expression is useful since the semi-infinite CMV matrix is better understood than the doubly infinite CMV matrix, due to the fact that the semi-infinite CMV matrix has a stronger connection to the theory of Jacobi operators and orthogonal polynomials on the real line. We will also suggest potential applications of our formula, in particular when analyzing orthogonal polynomials on the unit circle whose recurrence coefficients are Sturmian sequences.
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