Math 6420 • PDE

Let \(X\) be a separable Hilbert space and \((e_n)\) be a total orthonormal sequence of \(X\).
Let \(T:X\to X\) be a bounded linear operator satisfying
\[
  \sum_{n=1}^{\infty}\lVert Te_n\rVert^2<\infty.
  \tag{2}
\]
Define the operators \(T_k:X\to X\), \(k=1,2,\ldots\), that applied to an arbitrary
\[
  x=\sum_{n=1}^{\infty}x_ne_n\in X
  \qquad\text{give}\qquad
  T_kx=\sum_{n=1}^k x_nTe_n.
\]
\begin{enumerate}
\item[(a)] Are the operators \(T_k\) compact? Explain why or why not.

\item[(b)] Using equation (2), show that the sequence \((T_k)\) converges uniformly to \(T\),
i.e. show that \(\lVert T-T_k\rVert\to0\) as \(k\to\infty\).

\item[(c)] Is \(T\) compact? Explain why or why not.
\end{enumerate}