I have a bounded operator $T$ from $L^p$ to itself for $1 \leqslant p \leqslant \infty$. Furthermore, on $L^2$ we have that $T$ is self-adjoint.
Now I wish to relate $\|(Tf)g\|_{L^1}$ to $\|f(Tg)\|_{L^1}$ (equal up to a constant perhaps). What properties should I need for $T$ for this to hold?
The question is not really well-defined, but I don't know what property I should look for in my operator.