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Homework Statement
Consider the operator [tex]T:f(x)\rightarrow f(g(x))[/tex], where [tex]g:R\rightarrow R[/tex] is continuously differentiable and bijective. What is the adjoint of T?
Homework Equations
The definition of the adjoint is [tex]\langle f\mid T^{\dagger}\mid g\rangle=(T\mid f\rangle)^\dagger\mid g\rangle[/tex] for all [tex]g[/tex] in the domain. The domain is [tex]L^2(R)[/tex].
The Attempt at a Solution
I think the answer is [tex]T^\dagger:f(x)\rightarrow |h'(x)|f(h(x))[/tex], where [tex]h(x)[/tex] is the inverse function to [tex]g[/tex], so that [tex]h(g(x))=g(h(x))=x[/tex]. I'm not sure how to get this answer.