Trace formulas and Borg-type theorems for matrix-valued Jacobi and Dirac finite difference operators

Borg-type uniqueness theorems for matrix-valued Jacobi operators H and supersymmetric Dirac difference operators D are proved. More precisely, assuming reflectionless matrix coefficients A, B in the self-adjoint Jacobi operator H=AS(+)+A(-)S(-)+B (with S-+/- the right/left shift operators on the lattice Z) and the spectrum of H to be a compact interval [E-, E+], E- < E+, we prove that A and B are certain multiples of the identity matrix. An analogous result which, however, displays a certain novel nonuniqueness feature, is proved for supersymmetric self-adjoint Dirac difference operators D with spectrum given by [-E-+(1/2), -E--(1/2)] boolean OR [E--(1/2), E-+(1/2)], 0 <= E- < E+. Our approach is based on trace formulas and matrix-valued (exponential) Herglotz representation theorems. As a by-product of our techniques we obtain the extension of Flaschka's Borg-type result for periodic scalar Jacobi operators to the class of reflectionless matrix-valued Jacobi operators. (c) 2005 Elsevier Inc. All rights reserved.