Let $V$ be the space of complex $n$×$n$ matrices equipped with the inner product $\langle A,B\rangle=\operatorname{tr}(AB^*)$.Then let $L_B$,$R_B$ denote the linear operators on $V$ defined by $L_B(A)=BA$, $R_B(A)=AB$. Prove that if $B$ is an arbitrary fixed member of $V$, then $L_B$ is unitarily similar to $R_C$ where $C=B^T$.
I only know that normal operator is unitarily diagonalizable but in this question $B$ is arbitrary and I can't easily construct an unitary operator $U$ on $V$ s.t. $L_B=U^*R_CU$ is true. Even $\dim(V)=4$,
I still find the question not easy to solve.
Any help?
Sorry, the question is easy actually if you notice $U(BA)$=$U(A)B^T$ for any $A$ and guess $U(A)=A^T$ then verify that is true...