I would like to find the adjoint operator in the Hilbertspace $L^2(0,\infty)$ of the operator $$ (Ax)(t)=x(at), x\in L^2(0,\infty), a>0. $$
My calculation is the following; I use the substitution $u=at$.
$$ \langle Ax,y\rangle_{L^2}=\int\limits_0^{\infty}x(at)\overline{y(t)}\, dt=\int\limits_{0}^{\infty}x(u)\overline{\frac{1}{a}y\left(\frac{u}{a}\right)}\, du $$
So the adjoint operator $A^*\colon L^2(0,\infty)\to L^2(0,\infty)$ should be given by
$$ (A^*x)(u)=\frac{1}{a}x\left(\frac{u}{a}\right). $$