Let $S$ :$\Bbb C^3 \to \Bbb C^3$ defined by $S(z_1,z_2,z_3)=(2z_2+iz_3,iz_1,z_2)$. Use the definition of self-adjoint to prove S is self-adjoint.
So basically the definition is about $\langle T(x),y\rangle = \langle x,T(y)\rangle,$ but we also know S is self-adjoint if and only if the standard matrix rep of S is Hermitian. From here, I don't think this S is Hermitian...
Any thoughts or hints? Thanks in advance.