Consider the following condition on a bounded operator $T$ on a Hilbert space:

$\ \ \ \ \ $(A) there exists an orthonormal basis $(e_j)$ with $\sum_j\parallel Te_j\parallel<\infty$.

We have the implications

$\ \ \ \ \ $(trace class) $\ \Rightarrow\ $ (A) $\ \Rightarrow\ $ (Hilbert-Schmidt)

But what about the converse directions? Are they both false or is one of them true?