I have Legendre's equation
$$L(f)=\frac{1}{\sin(\theta)} \left(- \frac{d}{d\theta} \sin(\theta) \frac{df}{d \theta} \right)$$
Now I know that after substituting $\cos(\theta) =x$ we get a self-adjoint operator $T_Lg = (1-x^2)g''- 2xg' $ with domain $\operatorname{dom}T_L=\left\{g \in L^2(-1,1); g \in AC^1 (-1,1) , T_Lg \in L^2(-1,1); \lim(1-x^2)g'(x)=0 \right\}.$
Now, if I define an operator $G(f):=L(\frac{1}{\sqrt{\sin}}f)$ and I would like to know: What is the appropriate domain for $T_G$ and is this operator also self-adjoint?
If there is anything unclear, please comment on this question.