Hilbert–Schmidt operator in the Shale paper

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 $\{e_\alpha\}$ .

Remarks

Key properties (paper use):

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

Examples

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