Suppose V is a complex inner-product space and T ∈ L(V) is a
normal operator such that T9 = T8. Prove that T is self-adjoint
and T2 = T.
The Attempt at a Solution
Consider T9=T8. Now "factor out" T7 on both sides to get T7T2 =TT7. Now we represent T as a matrix. Since T is normal, it is diagonizable. Therefore it is invertible. Now T2=TT7T-7 so T2=(T)(I)7=T.