find the hermitian operator of an operator which attaches a matrix and its iverted

Question

$V=M_n\left(\mathbb{C}\right)$ with the standard inner product $<A,B>=tr\left(B^{*}A\right)$ and let $P\in V$ be an invertible matrix. We define an operator $T_p:V->V$ such that $$T_p\left(A\right)=P^{-1}AP$$

I need to find $adj(T_p)$

I tried by definition and it didn't workout too much for me, and also if I could express $T_p$ in the standard base, and then use $\left[T^{*}\right]_B=\left[T_B\right]^{*}$, but I don't see how to express $T_p$ in such way, so I'm stuck.

Answer 1

Hint: if $A, B\in V$, then $tr(A B) = tr(B A)$

Using this rule, try to find a matrix $C$ depending on $B$ such that \begin{equation} \langle T_p(A), B\rangle = \langle A, C\rangle \end{equation}