Weak convergence = norm convergence for trace class operators?

Question

Given a (separable) Hilbertspace $H$, I look at the traceclass operators $\mathfrak{S}_1$. I recall the fact that the weak convergence implies norm convergence in the sequence space $\mathcal{l}^1$. Does this also hold for $\mathfrak{S}_1$ which is $$\text{tr }(|A-A_j|)\to 0 \text{ if and only if } \text{tr }K(A-A_j)\to 0\; \forall K\in B(H) ?$$