The discussion focuses on the proof that the operator \( T^*T - TT^* \) is self-adjoint, utilizing properties of adjoint operators in linear algebra. The key steps involve recognizing that the adjoint of \( T^*T - TT^* \) simplifies to itself due to the identity \( T^{**} = T \). The confusion arose from misapplying the adjoint property \( (AB)^* = B^*A^* \), which was clarified in the conversation.
PREREQUISITESStudents and professionals in mathematics, particularly those studying linear algebra, functional analysis, or operator theory, will benefit from this discussion.
HallsofIvy said:(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.