Math 6710 • Applied Linear Operators and Spectral Methods

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
Prove that \(\|L\| = 1\).


\item
Explain why if \(\lambda \in \mathbb{C}\) with \(|\lambda| > 1\), then \(\lambda \in \rho(L)\), the resolvent set of \(L\).


\item
Prove that that if \(\lambda \in \mathbb{C}\) with \(|\lambda| < 1\), then \(\lambda \in \sigma_p(L)\), the point spectrum of \(L\).


\item
Deduce what is \(\sigma(L)\), the spectrum of \(L\).

\end{enumerate}