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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
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