For any $r\in [0,1]$ we give an example of a rigid operator whose spectrum is the annulus $\{\lambda\in \mathbb{C} : r \le |\lambda| \le 1 \} $ . In particular, when $r=0$ this operator is rigid and non-invertible, and when $r\in {\kern1pt}] 0,1 [ $ this operator is invertible but its inverse is not rigid. This answers two questions of Costakis, Manoussos and Parissis [Recurrent linear operators. Complex Anal. Oper. Theory 8 (2014), 1601–1643].