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 $\displaystyle\sum\_{m,n}|\langle Te\_n,e\_m\rangle|^p<+\infty$?



Schatten $p$-classes for small $p$
This is an elaboration of Christian Remling's comment above.
The answer is no. Not even for a finite rank operator. Take $\cal H$ to be $l^2({\mathbb N})$ and choose any sequence $(a\_n)$ in $l^2({\mathbb N})$ which is not in $l^p({\mathbb N})$. Let $(e\_n)$ be the usual standard ONB. Let $T(e\_1)=(a\_n)$ and $T(e\_j)=0$ for $j>1$. This operator has rank one. But
$$
\sum\_{n,m}|\langle Te\_n,e\_m\rangle|^p=\sum\_{m}|a\_m|^p=\infty.
$$


