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\chapter[Operators with Compact Resolvent]%
{Operators with Compact Resolvent\\
Which Are Close to Being Normal}

\section{Auxiliary propositions from function theory}
Here we give statements of known results from function theory
which are needed in what follows.

If $U$ is a domain in the complex plane $\bold C$ and the function
$\psi(z)$ is holomorphic in $U$, then let $M_\psi(U)=\sup\{|\psi
(z)|\colon z\in U\}$, and denote by $n_\psi(U)$ the number of roots of $\psi(z)$
in $U$ (counting multiplicity). Also, let $D_r=\{z\colon |z|&lt;r\}$, $M_\psi(r)=M
_\psi(D_r)$, and $n_\psi(r)=n_\psi(D_r)$.
\begin{lem}[Phragm\'en-Lindel\"of theorem] Suppose the
 function
$f(z)$ is holomorphic inside the angle $\Omega=\{z\colon | \arg
z|&lt;\pi(2\alpha)^{-1}\}$
$(\alpha\geq1)$ and on its sides, and for some
$\beta&lt;\alpha$ 
\begin{equation}
\varliminf_{r\to\infty}r^{-\beta}\log\sup_{|z|=r}|f(z)|&lt;\infty
\end{equation}

If $|f(z)|\leq M(|\arg z|=\pi(2\alpha)^{-1})$, then
$|f(z)|\leq M$ for all $z\in\Omega$.
\end{lem}

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