Math 6710 • Applied Linear Operators and Spectral Methods

Assume a bounded, self-adjoint operator \(T : H \to H,\) where \(H\) is a Hilbert space,
satisfies

\begin{equation}
\langle T x, x\rangle \ge \beta \|x\|^2,
\tag{1}
\end{equation}

for some \(\beta > 0\). The goal of this problem is to show that \(T\) is one-to-one and onto.
\begin{enumerate}

\item
Show that the nullspace of \(T\) is trivial, i.e. \(\mathcal{N}(T) = \{0\}\).


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


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


\item
Why do we have \(\mathcal{R}(T) = H\)?

\end{enumerate}