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

**boboYO** · Nov22-09, 08:12 PM

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** · Nov22-09, 10:17 PM

(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** · Nov23-09, 12:49 AM

Originally Posted by 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.

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