  This note finds a new characterization of complete Nevanlinna-Pick kernels on
the Euclidean unit ball.
  The classical theory of Sz.-Nagy and Foias about the characteristic function
is extended in this note to a commuting tuple $\bfT$ of bounded operators
satisfying the natural positivity condition of $1/k$-contractivity for an
irreducible unitarily invariant complete Nevanlinna-Pick kernel. The
characteristic function is a multiplier from $H_k \otimes \cE$ to $H_k \otimes
\cF$, {\em factoring} a certain positive operator, for suitable Hilbert spaces
$\cE$ and $\cF$ depending on $\bfT$. There is a converse, which roughly says
that if a kernel $k$ {\em admits} a characteristic function, then it has to be
an irreducible unitarily invariant complete Nevanlinna-Pick kernel. The
characterization explains, among other things, why in the literature an
analogue of the characteristic function for a Bergman contraction
($1/k$-contraction where $k$ is the Bergman kernel), when viewed as a
multiplier between two vector valued reproducing kernel Hilbert spaces,
requires a different (vector valued) reproducing kernel Hilbert space as the
domain.
