  A well-known theorem due to R. E. Curto and N. Salinas gives a necessary and
sufficient condition for the unitary equivalence of commuting tuples of bounded
linear operators acting on reproducing kernel Hilbert spaces. Inspired by this
theorem, we obtain a different but equivalent criterion for the unitary
equivalence of operators acting on reproducing kernel Hilbert spaces. As an
application, we describe the structure of intertwining operator and prove that
the decomposition of Cowen-Douglas operators is unique up to unitary
equivalence.
