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Mathematics Qualifying Exam Question Bank

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Question

Math 6420 • PDE

Fall 2015 • Question 1

Question

Math 6420 • PDE Fall 2015 • Question 1

Assume a bounded, self-adjoint operator \(T:H\to H\), where \(H\) is a Hilbert space, satisfies \[ \langle Tx,x\rangle \ge \beta \lVert x\rVert ^2, \tag {1} \] for some \(\beta >0\). The goal of this problem is to show that \(T\) is one-to-one and onto.

  1. Show that the nullspace of \(T\) is trivial, i.e. \(\mathcal {N}(T)=\{0\}\).
  2. Show that the range \(\mathcal {R}(T)\) is closed.
  3. Show that \(\mathcal {R}(T)^\perp =\{0\}\).
  4. Why do we have \(\mathcal {R}(T)=H\)?

Raw LaTeX

Assume a bounded, self-adjoint operator \(T:H\to H\), where \(H\) is a Hilbert space, satisfies
\[
  \langle Tx,x\rangle\ge\beta\lVert x\rVert^2,
  \tag{1}
\]
for some \(\beta>0\). The goal of this problem is to show that \(T\) is one-to-one and onto.
\begin{enumerate}
\item[(a)] Show that the nullspace of \(T\) is trivial, i.e. \(\mathcal{N}(T)=\{0\}\).

\item[(b)] Show that the range \(\mathcal{R}(T)\) is closed.

\item[(c)] Show that \(\mathcal{R}(T)^\perp=\{0\}\).

\item[(d)] Why do we have \(\mathcal{R}(T)=H\)?
\end{enumerate}

Topic Tags

Adjoint Operators and Riesz Representation Hilbert Spaces and Orthogonality Linear Programming and Optimization Self-Adjoint and Positive Operators