Let $H$ be a separable complex Hilbert space and $T\in\mathcal{B}(H)$. Let $\{v_i\}_{i\in\mathbb{N}}$ be an orthonormal basis for $H$. Then $T$ is of trace class if $$\sum_{i\in\mathbb{N}}\langle |T|v_i,v_i\rangle_H<\infty,$$ where $|T|$ is defined to be $\sqrt{T^*T}$. For such an operator, its trace is defined to be $$\text{tr}(T):=\sum_{i\in\mathbb{N}}\langle Tv_i,v_i\rangle_H,$$ which one can show is an absolutely convergent series.
Question: Suppose the series $$\sum_{i\in\mathbb{N}}\langle Tv_i,v_i\rangle_H$$ is absolutely convergent for every orthonormal basis $\{v_i\}_{i\in\mathbb{N}}$. Then is $T$ of trace class?
