Trace formulas and a Borg‐type theorem for CMV operators with matrix‐valued coefficients

Abstract

Abstract. We prove a general Borg-type inverse spectral result for a reflectionless unitary CMV operator (CMV for Cantero, Moral, and Velázquez [13]) associated with matrix-valued Verblunsky coefficients. More precisely, we find an explicit formula for the Verblunsky coefficients of a reflectionless CMV matrix whose spectrum consists of a connected arc on the unit circle. This extends a recent result [39] for CMV operators with scalar-valued coefficients.In the course of deriving the Borg-type result we also use exponential Herglotz representations of Caratheodory matrix-valued functions to prove an infinite sequence of trace formulas connected with CMV operators.

Spectral theory as influenced by Fritz Gesztesy

“…For further information on these circle of ideas see [16,17,37,44,45,46,47,50,51,52,53,55,59,69,107,111,112,113,124,132,145]. In particular, we mention also the review by Fritz [38].…”

Minimal Rank Decoupling of Full-Lattice CMV Operators with Scalar- and Matrix-Valued Verblunsky Coefficients

“…Next we present formulas for different unitary γ ∈ C m×m , linking various spectral theoretic objects associated with half-lattice CMV operators U (γ) ±,k0 . For the special case when γ = I m , these objects, and the relationships described below, have proven exceptionally useful (see, e.g., [19], [20], [49], [50], and [87]).…”

“…The relevance of this unitary operator U on ℓ 2 (Z) m , more precisely, the relevance of the corresponding half-lattice CMV operator U +,0 in ℓ 2 (N 0 ) m is derived from its intimate relationship with the trigonometric moment problem and hence with finite measures on the unit circle ∂ D. (Here N 0 = N ∪ {0}.) Following [19], [20], [49], [50], and [87], this will be reviewed in some detail, and also extended in certain respects, in Sections 2 and 3, as this material is of fundamental importance to the principal topics (such as decoupling of full-lattice CMV operators into direct sums of left and right half-lattice CMV operators and a similar result for associated Green's functions) discussed in this paper, but we also refer to the monumental two-volume treatise by Simon [71] (see also [70] and [72]) and the exhaustive bibliography therein. For classical results on orthogonal polynomials on the unit circle we refer, for instance, to [6], [41]- [43], [53], [75]- [77], [80], [81].…”

“…For classical results on orthogonal polynomials on the unit circle we refer, for instance, to [6], [41]- [43], [53], [75]- [77], [80], [81]. More recent references relevant to the spectral theoretic content of this paper are [22], [38]- [40], [49], [50], [51], [66], [69], and [87]. The full-lattice CMV operators U on Z are closely related to an important, and only recently intensively studied, completely integrable nonabelian version of the defocusing nonlinear Schrödinger equation (continuous in time but discrete in space), a special case of the Ablowitz-Ladik system.…”