Let $M$ be an almost complex manifold with almost complex structure $J$. We have that $J : TM \to TM$ is a vector bundle isomorphism. Naturally, this induces a map on sections, and with abuse of notation we denote by $J: \Gamma (TM) \to \Gamma (TM)$.
Now, a differential operator between vector bundles $E$ and $F$ over $M$ is a smooth map $ D : \Gamma (E) \to \Gamma (F)$ satisfying certain local properties such as locally being a polynomial expression of partial derivatives, and this gives it a degree.
My question is, is $J$ as expressed above in terms of sections a differential operator? And if so, what is its degree? I'm not quite sure how to go about to formally show this besides hand-wavy intuition.
I know $J$ can also be expressed as a $(1,1)$-tensor field. Would this give it degree $2$ if it is a differential operator?