Hilbert–Schmidt Operator

A bounded operator $X$ on a Hilbert space is Hilbert–Schmidt if $\|X\|_2^2=\sum_\alpha \|Xe_\alpha\|^2<\infty$ for some (hence any) orthonormal basis $\{e_\alpha\}$.

Key properties (paper use):

• Hilbert–Schmidt operators are compact .
• “Restricted” conditions are of the form $|T|-I\in HS$ (membership in $GL(H)_2$, $rGL(H)$, $rSp(K)$).

Example: On $\ell^2$, the diagonal operator $\mathrm{diag}(a_n)$ is HS iff $\sum_n |a_n|^2<\infty$.