Let $H$ be a separable Hilbert space with a fixed orthonormal basis $\{e_n\}_n$. For a bounded operator $T$ on $H$, the *diagonal* of $T$ is the unique operator $D_T$ on $H$ which is diagonal with respect to the above basis, and whose diagonal entries are given by $d_n=\langle T(e_n),e_n\rangle$. It is well know that if $T$ is positive and $D_T=0$, then necessarily $T=0$.

Question: If $T$ is positive and $D_T$ is compact, is $T$ necessarily compact?