If $H$ is an infinite dimensional separable hilbert space, define $T(H)$ as
$$A \in T(H) \qquad \text{iff} \qquad (A^*A)^{\frac{1}{2}} \, \, \text{bounded and compact} \, \, $$
and $$\sum_{i} \lambda_i((A^*A)^{\frac{1}{2}}) <\infty$$
where $\lambda_i$ are the eigenvalues with repeated multiplicity.
So clearly $(A^*A)^{\frac{1}{2}}$ is Hilbert Schmidt by a simple norm argument. Since H-S operators form a two sided ideal, how can I use this to show $T(H)$ is also a two sided ideal (in bounded operators)?