Notation for composition of partial derivative operator with "multiplication by function"

Question

Let's say I have an equation $\hat L u = u$, where $u$ is a function $x$ and $\hat L$ is the linear operator defined as $$ \hat L u(x) = f(x) u(x) + \partial_x (g(x) u(x)) + \partial_x (g(x) (\partial_x u(x))) $$

How would you write $\hat L$? One could write it as $$ \hat L = f(x) + \partial_x g(x) + \partial_x (g(x) \partial_x), $$ and the interpretation of the first and last terms are pretty straight forward, but the second term could be interpreted as the operator $$ \hat L u(x) = f(x) u(x) + \partial_x (g(x)) \cdot u(x) + \partial_x (g(x) (\partial_x u(x))) $$ Is there a better notation? Maybe like this? $$ \hat L = f(x) + \partial_x \circ g(x) + \partial_x \circ g(x) \circ \partial_x $$

Answer 1

You could perhaps use a dot symbol (\boldsymbol{\cdot}) \begin{equation} \hat{L} = f(x)\mathord{\boldsymbol{\cdot}} +\partial_x(g(x)\boldsymbol{\cdot}) + \partial_x(g(x)\partial_x\boldsymbol{\cdot}) \end{equation}