Let $T:H\to H$ be a compact operator on a complex Hilbert space.
Assume that
$$
\sup\_{(e\_j)}\sum\_j\left|\langle Te\_j,e\_j\rangle\right|<\infty,
$$
where the supremum extends over all orthonormal bases of $H$.
Does it follow that $T$ is a trace class operator?



Is this operator trace class?
Yes. Break the operator into it's real and imaginary parts: $A=\frac{1}{2}(T + T^\dagger)$ and $B=\frac{1}{2i}(T - T^\dagger)$. These are also compact and satisfy the same estimate. The estimate, applied to their respective eigenbases, shows that $A$ and $B$ are trace class. Thus $T=A + i B$ is also trace class.


