So I have the following task:
Let $T_j f(x)=\int_{\mathbb R}f(x-t)e^{it^3}\psi(2^{-j}t)\frac{dt}{t}$ for $j>0$. Prove that $\|T_j f\|_2\leq\|f\|_2$.
Now the hint suggests to use the $TT^*$ method. I am assuming that $T^*$ denotes the adjoint operator. But what exactly is the $TT^*$ method?
The $TT^*$ refers to a result that says that if one of
$$T,$$
or
$$T^*,$$
or
$$TT^*$$
(-and hence the name) is bounded then all of the above are bounded. A surface level google search didn't return that many results, but you can find statements in Stein,Shakarchi Vol IV, chapter 8 section 5(or 6?) Muscalu and Schlag...somewhere in the middle.
You can also show that the constants, i.e, the bound $k$ in $||T f|| \leq k ||f||$ , is the same (or at least related) in all cases. Furthermore - see the statement in Ginibre and Velo's paper here , which is the most general statement of the result I know of- we can use combinations of operators in context.
In your case, I suggest computing $T_jT_j^*$ and seeing if that is easier to bound, and if the constant that gives you the bound is $1$, as this is equivalent to what you are trying to show by the "$TT^*$ theorem". I'm not sure what context you are in and therefore what inner product you should compute in, but if the hint is to use $TT^*$ then this should at least be doable...