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Let V be a finite-dimensional inner product space. Let E be an operator on V such that E^2 = E. Show that E is self adjoint if and only if E*E = EE*.

Solution

If E is self adjoint, i.e. if E = E*, then apparently E*E = EE = E² = E and EE* = EE= E² = E so that E*E = EE*

Now let E*E = EE*. This means that E commutes with its adjoint, so that E is a normal operator. Then for all x, we have Ex= E* x. Similarly, IE is also a normal operator so that (IE)x = (IE*)x. Now, since (IE) Ex = 0, therefore (IE*) Ex = 0 so that E*E = E. In a similar way, since E (IE) x = 0, we have E*(IE)x = 0 so that E*E = E*. However, since E*E = E, we have E = E* i.e. E is self adjoint