Math 6420 • PDE

Consider the multiplication operator \(T:L^2[0,1]\to L^2[0,1]\) defined by
\(Tx(t)=tx(t)\).
\begin{enumerate}
\item[(a)] Show that \(T\) is a bounded, self-adjoint linear operator.
\item[(b)] Find the spectrum \(\sigma(T)\), point spectrum \(\sigma_p(T)\), continuous
spectrum \(\sigma_c(T)\), residual spectrum \(\sigma_r(T)\), and resolvent \(\rho(T)\).
\item[(c)] Show that the corresponding spectral family \((E_\lambda)_{\lambda\in\mathbb{R}}\)
is given by
\[
  E_\lambda x=
  \begin{cases}
    0, & \lambda<0,\\
    v_\lambda, & 0\le\lambda\le1,\\
    x, & \lambda>1,
  \end{cases}
  \qquad\text{where}\qquad
  v_\lambda(t)=
  \begin{cases}
    x(t), & 0\le t\le\lambda,\\
    0, & \lambda<t\le1.
  \end{cases}
\]
\end{enumerate}