  Let $Z(Ann(r,R))$ be the class of all continuous functions $f$ on the annulus
$Ann(r,R)$ in $\mathbb C^n$ with twisted spherical mean $f \times \mu_s(z)=0,$
whenever $z\in \mathbb C^n$ and $s >0$ satisfy the condition that the sphere
$S_s(z)\subseteq Ann(r, R) $ and ball $B_r(0)\subseteq B_s(z).$ In this paper,
we give a characterization for functions in $Z(Ann(r,R))$ in terms of their
spherical harmonic coefficients. We also prove support theorems for the twisted
spherical means in $\mathbb C^n$ which improve some of the earlier results.
