I have a linear operator in a Hilbert Space $H=L^{2}(0,1)$ that is $Tf(x)=xf(1-x^{3})$ and I want to find $T^{*}$, $T^{*}T$ and $\left \| T \right \|$. What should I do?
To find $T^*$, remember that a linear functional $l$ in $L^2(0,1)$ is given by a function $g\in L^2(0,1)$: $l(f) = \int_{(0,1)}f(x)g(x)\lambda(dx)$.