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?