- #1
GSpeight
- 31
- 0
Hi there,
Can anyone give me an hint/idea of how to prove Hilbert-Schmidt operators are compact? More specifically, if X is a seperable Hilbert space and T:X->X is a linear operator such that there exists an orthonormal basis [itex](e_{n})[/itex] such that [itex]\sum_{n} ||T(e_{n})||^{2}<\infty[/itex] then show that T is compact.
It looks like an easy exercise given that both definitions are given in terms of sequences but I'm being quite stupid so I'm having trouble.
Thanks for any help.
Can anyone give me an hint/idea of how to prove Hilbert-Schmidt operators are compact? More specifically, if X is a seperable Hilbert space and T:X->X is a linear operator such that there exists an orthonormal basis [itex](e_{n})[/itex] such that [itex]\sum_{n} ||T(e_{n})||^{2}<\infty[/itex] then show that T is compact.
It looks like an easy exercise given that both definitions are given in terms of sequences but I'm being quite stupid so I'm having trouble.
Thanks for any help.