Math 6420 • PDE

Let \(X,Y\) be Hilbert spaces and let \((e_n)\subset X\) be a total orthonormal sequence.
Recall that an operator \(T:X\to Y\) is said to be Hilbert-Schmidt if the Hilbert-Schmidt
norm
\[
  \lVert T\rVert_{\mathrm{HS}}:=\left(\sum_{n=1}^{\infty}\lVert Te_n\rVert^2\right)^{1/2}
\]
is finite.
\begin{enumerate}
\item[(a)] Show that the Hilbert-Schmidt norm is independent of the sequence \((e_n)\)
chosen. Hint: Use Parseval's identity.
\item[(b)] Show that a Hilbert-Schmidt operator is compact.
\item[(c)] For a Hilbert-Schmidt operator, \(T\), show that
\(\lVert T\rVert\le \lVert T\rVert_{\mathrm{HS}}\), where \(\lVert\cdot\rVert\) denotes the
operator norm.
\end{enumerate}