We can assume $T=0$. Let $H$ be a HIlbert space with orthonormal basis $(e_1,e_2,\dots)$. For $j\in{\mathbb N}$ let $T_j:H\to H$ be given by $T_j e_k=\delta_{j,k}e_k$ (Kronecker-delta). Then $T_j$ is trace class of trace norm one, but the sequence $T_j$ tends to zero strongly (and hence weakly).

Also, insisting $tr(T_j)\to 0$ does not help for you can replace $T_j$ with $S_j$ where $S_je_k=e_k$ if $k=j$, $-e_k$ if $k=j+1$ and zero otherwise.

We can assume $T=0$. Let $H$ be a HIlbert space with orthonormal basis $(e_1,e_2,\dots)$. For $j\in{\mathbb N}$ let $T_j:H\to H$ be given by $T_j e_k=\delta_{j,k}e_k$ (Kronecker-delta). Then $T_j$ is trace class of trace norm one, but the sequence $T_j$ tends to zero strongly (and hence weakly).

Also, insisting $tr(T_j)\to 0$ does not help for you can replace $T_j$ with $S_j$ where $S_je_k=e_k$ if $k=j$, $S_je_k=-e_k$ if $k=j+1$ and zero otherwise.