A bounded linear operator $T$ on a Hilbert space $\mathcal{H}$ is concave if, for each $x\in\mathcal{H}$, $\|T^2x\|^2-2\|Tx\|^2 +\|x\|^2 \leq 0$. In this paper, it is shown that if $T$ is a concave operator then so is every power of $T$. Moreover, we investigate the concavity of shift operators. Furthermore, we obtain necessary and sufficient conditions for N-supercyclicity of co-concave operators. Finally, we establish necessary and sufficient conditions for the left and right multiplications to be concave on the Hilbert-Schmidt class.