Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Adjoint of the scalar Lie derivative?

  1. Feb 25, 2010 #1
    For every continuous linear operator [tex]A: H \rightarrow H[/tex] from a Hilbert space [tex]H[/tex] to itself, there is a unique continuous linear operator [tex]A^*[/tex] called its Hermitian adjoint such that

    [tex]\langle Ax, y \rangle = \langle x, A^* y \rangle[/tex]​

    for all [tex]x,y \in H[/tex].

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

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

    [tex]\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,[/tex]​

    but it does not appear that [tex]\mathcal{L}_X[/tex] is self-adjoint.
    Last edited: Feb 25, 2010
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted