(1) $U(x)=a \langle x,b \rangle +b \langle x,a \rangle $, $a,b\in H\setminus \{0\}$. U is an operator from H to H and a,b are orthogonal elements. I want to calculate $||U||$
For this one I tried the following:
$||U(x)||^2=||a \langle x,b \rangle +b \langle x,a \rangle ||^2=| \langle x,b \rangle |^2||a||^2+| \langle x,a \rangle |^2||b||^2\le2||a^2|| |b||^2 ||x||^2$, by using Cauchy Schwarz
=> $||U||\le \sqrt{2}||a|| ||b||$ ? When do I have equality here and how can I derive $||U||=$ from it?
(2)$U:L^2([0,\pi])->L^2([0,\pi])$ defined by $U f(x)=\sin(x)\int_{0}^{\pi}f(t)\cos(t)dt+\cos(x)\int_{0}^{\pi}f(t)\sin(t)dt$
Here I have no idea how to derive $||U||$