Math 6420 • PDE

Consider the left shift operator \(L:\ell^2\to\ell^2\), defined for a sequence
\((x_n)\in\ell^2\) by
\[
  L(x_1,x_2,x_3,\ldots)=(x_2,x_3,x_4,\ldots).
\]
\begin{enumerate}
\item[(a)] Prove that \(\lVert L\rVert=1\).
\item[(b)] Explain why if \(\lambda\in\mathbb{C}\) with \(\lvert\lambda\rvert>1\), then
\(\lambda\in\rho(L)\), the resolvent set of \(L\).
\item[(c)] Prove that that if \(\lambda\in\mathbb{C}\) with \(\lvert\lambda\rvert<1\), then
\(\lambda\in\sigma_p(L)\), the point spectrum of \(L\).
\item[(d)] Deduce what is \(\sigma(L)\), the spectrum of \(L\).
\end{enumerate}