I work with the well known book of Dunford/Schwartz "Linear Operators (Part II)". At first I should mention that the general difference between self-adjoint and symmetric operators is obvious to me. But what's with "formally symmetric" and "formally self-adjoint" respective to a formal differential operator? Concerning the author a formal differential operator $\tau$ is given and he is called formally symmetric or formally self-adjoint if $\tau=\tau^*$. So it semed to me that it is almost the same and it won't be distinguished qualitatively.
While reading on, in some theorems one assumes "formally symmetric" formal differential operators on the one hand and on the other hand some theorems further one assumes "formally self-adjoint" formal differential operators. So to the readers view it seems to be a difference, otherwise I think one could generally assume the same (formally symmetric or formally self-adjoint).
So I only want resolve my comprehension of the correct meaning.
Formally symmetric operators act on real-valued functions (or extended real-valued) while formally self-adjoint operators act on complex-valued functions.