  For $s\in \mathbb R$ the weighted Besov space on the unit ball $\mathbb B_d$
of $\mathbb C^d$ is defined by $B^s_\omega=\{f\in \operatorname{Hol}(\mathbb
B_d): \int_{\mathbb B_d}|R^sf|^2 \omega dV<\infty\}.$ Here $R^s$ is a power of
the radial derivative operator $R= \sum_{i=1}^d z_i\frac{\partial}{\partial
z_i}$, $V$ denotes Lebesgue measure, and $\omega$ is a radial weight function
not supported on any ball of radius $< 1$.
  Our results imply that for all such weights $\omega$ and $\nu$, every bounded
column multiplication operator $B^s_\omega \to B^t_\nu \otimes \ell^2$ induces
a bounded row multiplier $B^s_\omega \otimes \ell^2 \to B^t_\nu$. Furthermore
we show that if a weight $\omega$ satisfies that for some $\alpha >-1$ the
ratio $\omega(z)/(1-|z|^2)^\alpha$ is nondecreasing for $t_0<|z|<1$, then
$B^s_\omega$ is a complete Pick space, whenever $s\ge (\alpha+d)/2$.
