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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2014 • Question 3

Question

Math 6420 • PDE Fall 2014 • Question 3

Assume a compact self-adjoint operator \(T:X\to X\), where \(X\) is a separable Hilbert space, satisfies \(\langle Tx,x\rangle \ge 0\) for all \(x\in X\).

  1. State the spectral theorem for compact self-adjoint operators.
  2. Show that all the eigenvalues of \(T\) are non-negative.
  3. Construct an operator \(S:X\to X\) such that \(T=S^2\).

Raw LaTeX

Assume a compact self-adjoint operator \(T:X\to X\), where \(X\) is a separable Hilbert space,
satisfies \(\langle Tx,x\rangle\ge0\) for all \(x\in X\).
\begin{enumerate}
\item[(a)] State the spectral theorem for compact self-adjoint operators.

\item[(b)] Show that all the eigenvalues of \(T\) are non-negative.

\item[(c)] Construct an operator \(S:X\to X\) such that \(T=S^2\).
\end{enumerate}

Topic Tags

Eigenvalues, Spectral Radius, and Gershgorin Theorem Linear Programming and Optimization Spectral Decay and Principal Eigenvalues Spectral Theory of Operators