Let V be an inner product space and T:V->V a linear operator. Prove that if T is normal, then T and T* have the same kernel (T* is the adjoint of T).
The Attempt at a Solution
Let us assume x is in the kernel of T. Then, TT*x =T*Tx = T*0= 0. So T*x is in the kernel of T but this doesn't mean T*x = 0 and I get stuck here. Any help would be greatly appreciated.