  In this paper, we first develop a notion of dominated splitting for $\mathbb
M(2,\mathbb C)$-sequences and show it is a stable property under $\|\cdot
\|_\infty$-perturbation. Then we show an energy parameter belongs to the
spectrum of a Jacobi operator, possibly singular, if and only if the associated
Jacobi cocycle does not admit dominated splitting. This generalizes the results
obtained by the second author [Z] in the scenario of Schr\"odinger operators.
Finally, we consider dynamically defined Jacobi operators whose base dynamics
is only assumed to be topologically transitive. We show an energy parameter
belongs to the spectrum of the operator defined by the base point with a dense
orbit if and only if the dynamically defined Jacobi cocycle does not admit
dominated splitting. This includes the original Johnson's theorem obtained by
R. Johnson [J] for Sch\"odinger operators and the main theorem obtained by C.
Marx [Ma] for Jacobi operators as special cases.
