Math 6420 • PDE

The right-shift operator \(R:\ell^2\to\ell^2\) is defined by
\(R(\xi_1,\xi_2,\ldots)=(0,\xi_1,\ldots)\) and the left-shift operator
\(L:\ell^2\to\ell^2\) is defined by \(L(\xi_1,\xi_2,\ldots)=(\xi_2,\xi_3,\ldots)\).
\begin{enumerate}
\item[(a)] Show that \(L\) and \(R\) are adjoint to each other.
\item[(b)] Show that \(\lVert R\rVert=\lVert L\rVert=1\).
\item[(c)] Prove that the spectrum of \(L\) is given by
\[
  \sigma_p(L)=\{\lambda\in\mathbb{C}:\lvert\lambda\rvert<1\},\quad
  \sigma_c(L)=\{\lambda\in\mathbb{C}:\lvert\lambda\rvert=1\},\quad
  \sigma_r(L)=\emptyset.
\]
\item[(d)] Prove that the spectrum of \(R\) is given by
\[
  \sigma_p(R)=\emptyset,\quad
  \sigma_c(R)=\{\lambda\in\mathbb{C}:\lvert\lambda\rvert=1\},\quad
  \sigma_r(R)=\{\lambda\in\mathbb{C}:\lvert\lambda\rvert<1\}.
\]
\end{enumerate}