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\title{MAT 247 \\ Assignment 4 \\ Due Thursday February 10}



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\begin{enumerate}
\item (Axler 7.9) Prove that a normal operator on a complex inner product space is self-adjoint iff all of its eigenvalues are real.

\item (Axler 7.16) Give an example of an operator $ T $ on a inner product space $ V $ with a subspace $ W $ such that $ W $ is $ T$-invariant, but $ W^\perp $ is not $T$-invariant.

\item Let $ V $ be an inner product space and let $ W $ be a subspace.  We have $ V = W \oplus W^\perp $.  Define a linear operator $ T : V \rightarrow V $ by $ T(w + u) = w - u $ if $ w \in W $ and $ u \in W^\perp$.  Prove that $ T $ is an isometry and is self-adjoint.

\item Prove the converse to (3).  More precisely, suppose that $ V $ is an inner product space and $T : V \rightarrow V $ is a self-adjoint isometry.  Show that there exists a subspace $ W $ of $ V $ such that $ T(w + u) = w - u $, whenever $ w \in W $ and $ u \in W^\perp $.
\end{enumerate}

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