Operator is normal iff ||Tv||=||T*v||, simple proof help

[boboYO]

http://img690.imageshack.us/img690/8659/linearalg.jpg I am having trouble understanding 7.6 . Specifically, getting from the 2nd line to the first line. How do we know that T*T-TT* is self adjoint?

 

[HallsofIvy]

(AB)*= B*A* and (A*)*= A.

The adjoint of T*T-TT* is (T*T- TT*)*= (T*T)*- (TT*)*= (T*)(T**)- (T**)T*. And the adjoint is 'dual'- that is, T**= T so that becomes T*T- TT* again.

 

[boboYO]

(AB)*= B*A* and (A*)*= A.

The adjoint of T*T-TT* is (T*T- TT*)*=(T*T)*- (TT*)*= (T*)(T**)- (T**)T*. And the adjoint is 'dual'- that is, T**= T so that becomes T*T- TT* again.

Ahh thank you very much, I got confused and thought (AB)*=A*B*.