Let $T:L_2 [0,1] \rightarrow \mathbb{C}$ define by $$ \int_0^1 f\left( \frac{x}{2} \right) e^x dx$$ Is $T$ bounded? If $T$ is bounded compute $||T||$, Is $T$ compact? Compute the adjoint of T.
First, when I try to bound $||Tf||$ I'm not sure how work with $ f\left( \frac{x}{2} \right)$ for the compacity of $T$ I try to use the definition of compact operators but I don't sure to these is the right way.
Any hint or suggestion I will very grateful