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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., Comment: Published: Annales de l'institut Fourier, 59 no. 6 (2009), p. 2509-2523