Let $V$ be the space of $n \times n$ matrices over the complex numbers, with the inner product $(A,B) = tr (AB^*)$. Let $Ρ$ be a fixed invertible matrix in $V$, and let $Τ_P$ be the linear operator on $V$ defined by $Τ_P (A) = Ρ^{-1}ΑΡ$. Find the adjoint of $Τ_P$.
If adjoint of $Τ_P$ exists then $(T_PA,B) = (A,T_P^*B)$ where $T_P^*$ is the adjoint.
By definition of the inner product on $V$, we have $$ \begin{align} tr(T_p(A) B^*)&=tr(P^{-1}AP B^*) \\ &=tr(APB^* P^{-1}) \ \ \textrm{ (since $tr(AB)=tr(BA)$ for any $A,B$ in $V$) } \end{align} $$
Thus, the adjoint operator $T_P^*$ must satisfy $$ (T_P^*(B))^* = P B^* P^{-1}. $$ Then we have $$ T_P^*(B) = (P^{-1})^* B P^*. $$