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.