Let $H=(\mathbb R^2,(.,.))$ and $M=\{(x,0)|x\in\mathbb R\}, N=\{(x,x\tan(\theta)|x\in\mathbb R)$ with $\theta\in(0,\frac{\pi}{2})$.
Now I would like to find a $T_\theta\in B(H,H)$ with $T^2_\theta=T_\theta, T_\theta(H)=M$ and $Ker(T_\theta)=N$
$B(H,H)$ are the bounded linear operators.
What I am also intersted in is how to calculate $||T_\theta||$ then, but I have no idea finding the correct $T_\theta$