Suppose $\mathcal H$ is a separable Hilbert space and $T$ is a compact self-adjoint operator on $\mathcal H$. Let $\{e_n\}$ be an orthonormal basis for $\mathcal H$. Fix $1<p<2$.

Does $T\in$Schatten p-class imply that $\displaystyle\sum_{m,n}|\langle Te_n,e_m\rangle|^p<+\infty$