The trace operator is a map on tensor spaces
\begin{equation*} \tr\colon T^{(k+1,l+1)}(V)\to T^{(k,l)}(V), \end{equation*}which sum ups the components of the last covariant and contravariant indices, i.e.
\begin{equation*} (\tr F)^{i_1\ldots i_k}_{j_1\ldots j_l}= F^{i_1\ldots i_km}_{j_1\ldots j_lm}, \end{equation*}where we are using Einsteins sum convention .