Assume that $X$ is a non vanishing vector field on the torus $\mathbb{T}^2$.

We define two linear operators $T,S$ on the space of smooth functions on $\mathbb{T}^2$ as follows:

$T(f)=X.f$
$S(f)= *d(\alpha_{f})$ where $*$ is the Hodge operator and $\alpha_{f}$ is the $1\; \_$ form on the torus with $\alpha_{f}(h)=fX.h$ the later is based on a Riemannian metric on the torus.

Is $D=T^2+S^2$ an elliptic operator?If yes is its index depend on the vector field $X$ and a Riemannain metric on the torus?If the index depends on $X$, are there some dynamical interpretation for the index of this operator?