We have just covered Hilbert-Schmidt operators in class (which I missed) and I am having a hard time getting my head around them. I know the definition:
If $H$ is a Hilbert space and $T\in\mathcal{L}(H)$ then $T$ is Hilbert-Schmidt if there is some orthonormal basis of $H$ such that: $\sum_n \lVert T(e_n)\rVert^2<\infty$
So I am a bit unsure what this is saying about the operator? Is this saying that the operator is "bounded enough" in some sense, to do something?
Thanks very much for the help (sorry for my confusion and rubbish question)