Let $T$ be an operator on $L^2(0, 1)$ defined as $T[f(x)]=f(\frac{x}{2})$. Find $\|T\|$ and $T^*$. Does $TT^*=T^*T$ hold? Find also eigenvalues and eigenvectors of $T$.
My attempt for finding $\|T\|$ is the following. $$ \|Tf\|^2=\int_0^1|f(\frac{x}{2})|^2dx=2\int_0^{\frac{1}{2}}|f(y)|^2dy\leq 2\|f\|^2\Rightarrow\frac{\|Tf\|}{\|f\|}\leq\sqrt2 $$ How can I prove the equality? Moreover, some hints for the other questions?