1. The problem statement, all variables and given/known data Let V be a complex inner product space, and let T be a linear operator on V. Define T_1 = 1/2 (T + T*) and T_2 = (1/2i)(T – T*) a) Prove that T_1 and T_2 are self-adjoint and that T = T_1 + T_2 b) Suppose also that T = U_1 + U_2, where U_1 and U_2 are self-adjoint. Prove that U_1 = T_1 and U_2 = U_2. c) Prove that T is normal iff (T_1)(T_2) = (T_2)(T_1) 2. Relevant equations Self-adjoint: T = T* Normal: TT* = T*T 3. The attempt at a solution (a) was very easy. However, I cannot get started on (b) at all. For (c) One direction: assume (T_1)(T_2) = (T_2)(T_1), show T is normal Substitute: (1/2 (T + T*))((1/2i)(T – T*)) = ((1/2i)(T – T*))( 1/2 (T + T*)) (1/4i)(T^2 – T* ^2) = (1/4i)(T^2 – T*^2) I’m not sure what to do now, or if it’s even the right direction. Thanks for your help!