Let $\mathscr{A}_n$ be a sequence of trace-class operators on a Hilbert space $\mathcal{H}$ and let further $\mathscr{A}$ be another trace-class operator on the same space.
Assume that $\mathscr{A}_n$ converges to $\mathscr{A}$ in Hilbert-Schmidt norm. Does $\mathscr{A}_n$ converge to $\mathscr{A}$ in trace norm?
Best Answer
On $\ell^{2}$ if $T_n((a_k))=\left(0,0,..,0,\frac {a_n} n,...,\frac {a_{2n}}{2n},0,0...\right)$ then $T_n$ converges in HS norm to $0$ but not in trace norm.