  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^\pm 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}] \cup [E_-^{1/2},E_+^{1/2}], 0
\leq E_- < E_+.
