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Section 8.7 Exercises

Find the point spectrum and the corresponding eigenfunctions in \(L^2([0,\pi])\) for the operators \(A\) and \(B\) with
\begin{align*} A=-\frac{d^2}{dx^2},\amp\qquad\amp D(A)=\{f\in L^2(0,\pi); Af \in L^2(0,\pi), f(0)=0, f(\pi)=0\}, B=-\frac{d^2}{dx^2},\amp\qquad\amp D(B)=\{f\in L^2(0,\pi); Bf \in L^2(0,\pi), \frac{d}{dx}f(0)=0,\frac{d}{dx} f(\pi)=0\}. \end{align*}

Consider a linear operator \(A\) on a Banach space with adjoint \(A^*\text{.}\) Show that
\begin{equation*} \lambda \in \sigma_p(A) \qquad \Longrightarrow \qquad \bar \lambda \in \sigma_r(A^*) \cup \sigma_p(A^*). \end{equation*}

Consider the Hilbert space \(l^2\text{,}\) which consist of sequences \(x=(x_1, x_2, \dots)\) with bounded norm
\begin{equation*} \|x\|_2^2 = \sum_{i=1}^\infty |x_i| ^2. \end{equation*}
We define two operators, a left-shift \(L\) and a right-shift \(R\) as
\begin{equation*} L(x_1, x_2, x_3, \dots) = (x_2, x_3, \dots), \qquad R(x_1, x_2, x_3, \dots) = (0, x_1, x_2, x_3, \dots) . \end{equation*}
  1. Find the adjoints of \(L\) and \(R\text{.}\)
  2. \(\sigma(L), \sigma_p(L), \sigma_c(L), \sigma_r(L)\text{.}\)
    • Find the norm of \(L\) and use it to find the spectral radius of \(L\text{.}\) Define a ball that contains the spectrum \(\sigma(L)\text{.}\)
    • Find the point-spectrum of \(L\text{.}\)
    • Show that \(\sigma_r(L) = \emptyset\text{.}\)
    • Use the fact that the spectrum is a closed set to identify all three components.
  3. \(\sigma(R), \sigma_p(R), \sigma_c(R), \sigma_r(R)\text{.}\)
    • Find the norm of \(R\) , the spectral radius of \(R\text{,}\) and define a ball that contains the spectrum \(\sigma(R)\text{.}\)
    • Find the point-spectrum of \(R\text{.}\)
    • Use the previous results and the result from Exercise 22 to find the residual spectrum of \(R\text{.}\)
    • Find the continuous spectrum of \(R\text{.}\)
    • Use the fact that the spectrum is a closed set to identify all three components.

  1. Show that the operator on \(L^2(0,1)\) given by
    \begin{equation*} A=\frac{\partial}{\partial x} ,\qquad D(A)=\{u\in H^1([0,1]) ; u(0) = u(1) \} \end{equation*}
    is skew adjoint (\(A^*=-A, D(A^*)=D(A)\)).
  2. Show that the point spectrum of \(A\) is purely imaginary.