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## Homework Statement

Suppose V is a complex inner-product space and T ∈ L(V) is a

normal operator such that T

^{9}= T

^{8}. Prove that T is self-adjoint

and T

^{2}= T.

## Homework Equations

## The Attempt at a Solution

Consider T

^{9}=T

^{8}. Now "factor out" T

^{7}on both sides to get T

^{7}T

^{2}=TT

^{7}. Now we represent T as a matrix. Since T is normal, it is diagonizable. Therefore it is invertible. Now T

^{2}=TT

^{7}T

^{-7}so T

^{2}=(T)(I)

^{7}=T.