Adjoint of an Operator - Considerations and Solution

[qudit]

Homework Statement

Consider the operator $$T:f(x)\rightarrow f(g(x))$$, where $$g:R\rightarrow R$$ is continuously differentiable and bijective. What is the adjoint of T?

Homework Equations

The definition of the adjoint is $$\langle f\mid T^{\dagger}\mid g\rangle=(T\mid f\rangle)^\dagger\mid g\rangle$$ for all $$g$$ in the domain. The domain is $$L^2(R)$$.

The Attempt at a Solution

I think the answer is $$T^\dagger:f(x)\rightarrow |h'(x)|f(h(x))$$, where $$h(x)$$ is the inverse function to $$g$$, so that $$h(g(x))=g(h(x))=x$$. I'm not sure how to get this answer.

 

[arkajad]

Write down the conditions in 2) as integrals. Change the variables.