Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1994 • Question 16

Question

Math 6410 • ODE Fall 1994 • Question 16

Let \(T\) be a linear operator in a Hilbert space \(\mathcal {H}\) with domain \(D(T)\) dense in \(\mathcal {H}\). Complete the following:

Define the adjoint operator \(T^*\).
Let \(G(T)\) be the graph of \(T\) in \(\mathcal {H}\oplus \mathcal {H}\). Define the operator \(V\) on \(\mathcal {H}\oplus \mathcal {H}\) by \(V(f,g)=(g,-f)\). Prove that the graph of \(T^*\) is the orthogonal complement in \(\mathcal {H}\oplus \mathcal {H}\) of \(V(G(T))\), that is, \[ G(T^*)=(V(G(T)))^\perp . \]
Let \(\overline {G(T)}\) be the closure in \(\mathcal {H}\oplus \mathcal {H}\) of the graph of \(T\). Prove that \[ \mathcal {H}\oplus \mathcal {H}=V(\overline {G(T)})\oplus G(T^*). \]

Let \(T\) be a linear operator in a Hilbert space \(\calH\) with
domain \(D(T)\) dense in \(\calH\). Complete the following:

\begin{enumerate}
\item[(a)] Define the adjoint operator \(T^*\).

\item[(b)] Let \(G(T)\) be the graph of \(T\) in \(\calH\oplus\calH\). Define the operator
\(V\) on \(\calH\oplus\calH\) by \(V(f,g)=(g,-f)\). Prove that the graph of \(T^*\) is the
orthogonal complement in \(\calH\oplus\calH\) of \(V(G(T))\), that is,
\[
G(T^*)=(V(G(T)))^\perp.
\]

\item[(c)] Let \(\overline{G(T)}\) be the closure in \(\calH\oplus\calH\) of the graph of
\(T\). Prove that
\[
\calH\oplus\calH=V(\overline{G(T)})\oplus G(T^*).
\]
\end{enumerate}

Topic Tags

Adjoint Operators and Riesz Representation Bounded Linear Operators Hilbert Spaces and Orthogonality