Sum of $k$ smallest singular values

Question

The $k$th Ky Fan norm $\lVert\cdot\rVert_{(k)}$ is defined as the sum of the $k$ largest singular values. Furthermore, for an $m\times n$ matrix $A$ $$ \lVert A\rVert_{(k)} = \max_{UU^*=VV^*=I_k}|\text{tr}(UAV)|. $$ For Hermitian matrices the sum of the $k$ smallest eigenvalues, denoted $E_k(H)$, is $$ E_k(H) = \min_{UU^*=I_k}\text{tr}(UHU^*), $$ so I was wondering if the sum of the $k$ smallest singular values $S_k(A)$ for a matrix $A$ can be written $$ S_k(A) = \min_{UU^*=VV^*=I_k}|\text{tr}(UAV)|. $$ As I have not seen this expression I thought that perhaps it is not true, and that the argument used for the two previous examples fails at some point. Could anyone shed some light on this?