Let us consider the finite group algebra $\ell^1(\mathbb{Z}_n)$. Let $x=(x_0,\cdots,x_{n-1})$ in $\ell^1(\mathbb{Z}_n)$ and define $$M_x: \ell^1(\mathbb{Z}_n)\to \ell^1(\mathbb{Z}_n) : M_x(a)=a*x$$ where $*$ is just the convolution product.

Q. Any characterization (formula) for linear maps $T: \ell^1(\mathbb{Z}_n)\to \ell^1(\mathbb{Z}_n)$ satisfying $T^2=M_x$?