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Question

Math 6420 • PDE

Fall 2018 • Question 4

Question

Math 6420 • PDE Fall 2018 • Question 4

Consider the operator \(T:L^2[0,1]\to L^2[0,1]\) defined by \((Tx)(t)=tx(t)\).

  1. Show that \(T\) is self-adjoint and satisfies \(\langle x,Tx\rangle \ge 0\) for every \(x\in L^2[0,1]\).
  2. Find the spectrum \(\sigma (T)\), point spectrum \(\sigma _p(T)\), continuum spectrum \(\sigma _c(T)\), residual spectrum \(\sigma _r(T)\) and resolvent \(\rho (T)\).

Raw LaTeX

Consider the 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 self-adjoint and satisfies
\(\langle x,Tx\rangle\ge0\) for every \(x\in L^2[0,1]\).
\item[(b)] Find the spectrum \(\sigma(T)\), point spectrum \(\sigma_p(T)\), continuum
spectrum \(\sigma_c(T)\), residual spectrum \(\sigma_r(T)\) and resolvent \(\rho(T)\).
\end{enumerate}

Topic Tags

Adjoint Operators and Riesz Representation Dual Spaces and Bounded Linear Functionals Linear Programming and Optimization Point, Continuous, and Residual Spectrum