My linear algebra textbook gives the definition of the Adjoint Operator and then says,
You should verify the following properties:
- Additivity: $(S + T)^* = S^* + T^*$
- Conjugate homogeneity: $(aT)^* = \overline{a}\,T^*$
- Adjoint of adjoint: $(T^*)^* = T$
- Identity: $I^* = I$, where $I$ is the identity operator on $V$.
I've stared at the pages for a couple hours now. How do you verify this?
Here's my attempt at a proof for Adjoint of Adjoint: (T*)* = (T*v, w)* = (v, Tw)* = (Tv, w) = T
Is that correct reasoning?
BTW, this is NOT homework. Just reading for pleasure.
Thanks!