An operator $A$ on $\Bbb R^2$ for which $\langle Ax,x\rangle=0$ for all $x$ and $\|A\| = 1$.

Question

An operator $A$ on $\Bbb R^2$ for which $\langle Ax,x\rangle=0$ for all $x$ and $\|A\| = 1$.

Let $e_1 = (1,0), e_2 = (0,1)$. And take $A$ to be such that $Ae_1 = e_2, Ae_2 = e_1$.

Then we have $\langle Ax,x\rangle=0$ for all $x$ and $\|A\| = 1$.

Is the example correct?

Answer 1

This is not correct, because $\langle A(e_1+e_2),e_1+e_2\rangle=2$.

Take $A$ such that $Ae_1=e_2$ and $Ae_2=-e_1$ instead.