Adjoint of the scalar Lie derivative?

[fuzzytron]

For every continuous linear operator $$A: H \rightarrow H$$ from a Hilbert space $$H$$ to itself, there is a unique continuous linear operator $$A^*$$ called its Hermitian adjoint such that

$$\langle Ax, y \rangle = \langle x, A^* y \rangle$$​

for all $$x,y \in H$$.

Given that $$\mathcal{L}_X: \Omega^0(M) \rightarrow \Omega^0(M)$$ (i.e., the Lie derivative on differential 0-forms over a manifold $$M$$) is such an operator, what is its Hermitian adjoint?

Ultimately I'm after a "directional" analog of Green's first identity, something like

$$\int_M (\mathcal{L}_X f)^2 = \langle \mathcal{L}_X f, \mathcal{L}_X f \rangle = \langle f, \mathcal{L}_X ( \mathcal{L}_X f ) \rangle,$$​

but it does not appear that $$\mathcal{L}_X$$ is self-adjoint.

 

[pat_connell]

The Hermitian adjoint of \mathcal{L}_X: \Omega^0(M) \rightarrow \Omega^0(M) is the operator \mathcal{L}_X^*: \Omega^0(M) \rightarrow \Omega^0(M) defined by \langle \mathcal{L}_X f, g \rangle = \langle f, \mathcal{L}_X^* g \rangle for all f, g \in \Omega^0(M).