\documentclass[12pt]{amsart}
\usepackage{amsmath, amssymb}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{axiom}{Axiom}
\newtheorem{remark}{Remark}
\newtheorem{exercise}{Exercise}[section]
\theoremstyle{definition}
%\numberwithin{equation}{section}
%\newcommand{\thmref}[1]{Theorem~\ref{#1}}
%\newcommand{\secref}[1]{\S\ref{#1}}
%\newcommand{\lemref}[1]{Lemma~\ref{#1}}
\begin{document}
%\renewcommand{\baselinestretch}{2}
\title[Essential Norms on Planar Domains]{The Essential Norm of a Composition Operator on a Planar Domain}
\date{August 29, 1997}
\author{Stephen D. Fisher}
\author{Jonathan E. Shapiro}
\address{Department of Mathematics\\
Northwestern University}
\subjclass{Primary 47B38, Secondary 30H05, 46E20}
\maketitle
\begin{abstract}
We generalize to finitely connected planar domains the result of Joel
Shapiro which gives a formula for the essential norm of a composition
operator. In the process, we define and give some properties of a
generalization of the Nevanlinna counting function and prove generalizations
of the Littlewood inequality, the Littlewood--Paley identity, and change of
variable formulas, as well.
\end{abstract}
\small{\ }
\section{Introduction}
Let $\Omega $ be a domain in the plane. For $1\le p<\infty $, the Hardy
space $H^p=H^p(\Omega )$ is defined to be those analytic functions $f$ on $%
\Omega $ for which the subharmonic function $|f(z)|^p$ has a harmonic
majorant. Once we specify a base point $t_0\in \Omega $, we define the norm
of $f$ to be $p^{th}$ root of the value at $t_0$ of the (unique) least
harmonic majorant of $|f|^p$. A different choice of the base point gives an
equivalent norm on $H^p$; this is an application of Harnack's inequality.
The Hardy space $H^\infty $ is the space of bounded analytic functions on $%
\Omega $ with the supremum norm. For more on the Hardy spaces, see \cite{R},
\cite{SDF}.
An analytic function $\varphi $ that maps $\Omega $ into itself determines a
composition operator $C_\varphi $ on $H^p$ given by
\begin{equation}
C_\varphi f=f\circ \varphi .
\end{equation}
$C_\varphi $ is a bounded operator on $H^p$. One simple way to see this is
to note that if $u_f$ is the least harmonic majorant of $|f|^p$, then $%
u_f\circ \varphi $ is an harmonic majorant of $|f\circ \varphi |^p$ and so
\begin{equation*}
||f\circ \varphi ||^p\le u_f(\varphi (t_0))\le Ku_f(t_0)
\end{equation*}
where $K$ is a constant that, again by Harnack's inequality, depends only on
the domain $\Omega $, and the points $t_0$ and $\varphi (t_0)$.
\bigskip In this paper we are concerned with $H^p$ on a domain $\Omega $
that is finitely-connected; that is, has only a finite number of
complementary components. In this setting, it is known \cite{F} that $%
C_\varphi $ is compact on some $H^p,1\le p<\infty $, if and only if it is
compact on all $H^p$. We therefore concentrate on $C_\varphi $ acting on $%
H^2 $. The main result of this paper is an extension of the theorem of Joel
Shapiro \cite{JHS} on the essential norm --- distance to the set of compact
operators --- of the composition operator $C_\varphi $ that he proved when $%
\Omega $ is the unit disk. To understand the statement of Shapiro's theorem
we must first define the Nevanlinna counting function of $\varphi $.
\bigskip
\begin{definition}
Let $\Delta $ be the open unit disk and suppose that $\varphi $ is an
analytic function mapping $\Delta $ into itself. The Nevanlinna counting
function for $\varphi $ is, for $w\neq \varphi \left( 0\right) $,
\begin{equation}
N_\varphi (w)=\sum_{\varphi (z)=w}-\log |z|.
\end{equation}
\end{definition}
With this background, we can state Joel Shapiro's result. \bigskip
\begin{theorem}
Suppose that $\varphi $ is an analytic function that maps $\Delta $ into $%
\Delta $ with $\varphi (0)=0$. Let $\left\| C_\varphi \right\| _e$ denote
the essential norm of $C_\varphi $ as an operator on $H^2$. Then
\begin{equation*}
\left\| C_\varphi \right\| _e^2=\limsup_{\left| w\right| \rightarrow
1^{-}}\,\left[ \frac{N_\varphi (w)}{-\log |w|}\right] .
\end{equation*}
In particular, $C_\varphi $ is compact on $H^2$ if and only if
\begin{equation*}
\lim_{\left| w\right| \rightarrow 1^{-}}\frac{N_\varphi (w)}{-\log |w|}=0.
\end{equation*}
\end{theorem}
The development of this paper follows the arguments of Shapiro in \cite{JHS}
closely, altering several parts as necessary to allow for the change in
setting.
\section{Background\label{background}}
Let $D$ be a domain in the plane whose universal covering surface is the
open unit disc $\Delta $ and let $\Pi $ be the covering map. The \emph{%
Poincar\'{e} metric} for $D$ is defined at $\zeta =\Pi (z)\in D$ by
\begin{equation*}
\lambda _D(\zeta )=|\Pi ^{\prime }(z)|(1-|z|^2).
\end{equation*}
It is shown in \cite[p. 44]{IK} that the value of $\lambda _D(\zeta )$ is
independent of the particular choice of $z\in \Delta $ with $\Pi (z)=\zeta $.
\medskip
If $D$ is regular for the Dirichlet problem, we denote the Green's function
for $D$ with pole at $p\in D$ by $g_D(z;p)$. The domain $D$ is omitted
unless confusion is possible.
\medskip
In this paper we shall generally be concerned with a planar domain $\Omega $
whose complement consists of a finite number of disjoint non-trivial
continua. Such a domain is conformally equivalent to one whose boundary
consists of a finite number of disjoint analytic simple closed curves;
indeed, it is conformally equivalent to a domain whose boundary components
are circles. Since the conformal mapping gives an isometry of the
corresponding Hardy spaces, we may assume, and shall do so, that the
components $\Gamma _0,...,\Gamma _p$ of $\Gamma $ are circles, with $\Gamma
_0$ the boundary of the unbounded component of the complement of $\Omega $.
We let $\omega _{t_0}$ denote the harmonic measure on $\Gamma $ for the
(fixed) base point $t_0$. It is standard \cite{F} that each $H^2$ function $%
f $ on $\Omega $ has boundary values almost everywhere on $\Gamma $, that
these boundary values lie in $L^2(\Gamma ,\omega _{t_0})$, and that the
correspondence of $f$ to its boundary values is an isometry of $H^2$ onto a
closed subspace of $L^2(\Gamma ,\omega _{t_0})$. We let $\Omega _j$ be the
region outside $\Gamma _j,j=1,...,p$, including the point at $\infty $ and $%
\Omega _0$ be the region inside $\Gamma _0$. Each of the regions $\Omega _j$
is conformally equivalent to the unit disk $\Delta $ via a linear fractional
transformation. When we write $H^2(\Omega _j)$ for the Hardy space for this
region, we shall always assume that the norm is taken with respect to the
base point $t_0$.
\medskip
\subsection{Factorization of $H^p$ functions}
There is a factorization of functions in $H^p(\Omega )$, developed in \cite
{VZ}, that parallels that for $H^p$ functions on the unit disc. Here we give
a summary; additional details may be found in \cite[Section 4.7]{SDF}.
\medskip
Let $\mathcal{G}$ be the group of linear fractional transformations of $%
\Delta $ onto itself that leave the covering map $\Pi $ invariant: $\Pi
\circ \tau =\Pi ,\tau \in \mathcal{G}$. An analytic function $h$ on $\Delta $
is \textit{modulus automorphic} if for each $\tau \in \mathcal{G}$ there is
a unimodular constant $c=c(\tau )$ such that $h\circ \tau =ch$. Each modulus
automorphic function $h$ corresponds to a function $f$ on $\Omega $ by $%
h(z)=f(\Pi (z)),z\in \Delta $. The modulus of $f$ is single-valued, but $f$
itself has unimodular periods in the sense that analytic continuation of a
function element $(f,\mathcal{O})$ along any curve $\gamma $ in $\Omega $
leads to the function element $(cf,\mathcal{O})$, where $c$ is a unimodular
constant that depends only on the homotopy class of $\gamma $. The class of
such multiple-valued analytic functions with single-valued modulus whose $%
p^{th}$ power has a harmonic majorant will be denoted by $MH^p(\Omega )$.
\medskip
A \emph{Blaschke product} $B$ is an element of $MH^\infty (\Omega )$ with
\begin{equation*}
\log |B(z)|=-\sum_kg_\Omega (z;w_k),\hspace{.4cm}\sum_kg_\Omega
(w_k;t_0)<\infty .
\end{equation*}
If there are only a finite number of zeros, then the second condition is
automatically satisfied.
\medskip
A \emph{singular inner function} $S$ is an element of $MH^\infty $ with
\begin{equation*}
\log |S(z)|=-\int_\Gamma P(s;z)d\nu (s)
\end{equation*}
where $\nu $ is a non-negative Borel measure on $\Gamma $ that is singular
with respect to harmonic measure $\omega _{t_0}$ and $P(\cdot ;z)$ is the
Poisson kernel for $z\in \Omega $.
\medskip
An \emph{outer function} in $MH^p$ is an element $F$ of $MH^p$ of the form
\begin{equation*}
\log |F(z)|=\int_\Gamma u(s)P(s;z)d\omega _{t_0}(s)
\end{equation*}
where $u\in L^1(\Gamma ,\omega _{t_0})$ and $e^u\in L^p(\Gamma ,\omega
_{t_0})$.
\medskip
The basic theorem on factorization is this.
\begin{theorem}
\label{factorization}Each function $f\in MH^p(\Omega )$ has a factorization
as
\begin{equation*}
f=BSF
\end{equation*}
where $B$ is a Blaschke product, $S$ is a singular inner function, and $F$
is an outer function in $MH^p(\Omega )$. The factors are unique up to
multiplication by unimodular constant. Even if $f$ is single-valued, the
factors need not be.
\end{theorem}
\medskip
\subsection{The Nevanlinna counting function}
Our first goal is to generalize the Nevanlinna counting function to the
domain $\Omega $ and understand some of its properties.
\begin{definition}
Let $\varphi :\Omega \to \Omega $ be an analytic function. The Nevanlinna
counting function for $\varphi $, $N_\varphi (w)$ for $w\in \Omega \setminus
\{\varphi (t_0)\}$, is
\begin{equation*}
N_\varphi (w)=\sum_{\varphi (z)=w}g_\Omega (z;t_0).
\end{equation*}
\end{definition}
Note that this reduces to the counting function defined previously if $%
\Omega $ is the unit disk $\Delta $ and $t_0=0$.
For the Nevanlinna counting function on the unit disk, there is the
classical theorem of Littlewood \cite{L}:
\begin{theorem}
Let $\psi $ be a holomorphic self--map of the unit disk $\Delta $. Then
\begin{equation}
N_\psi (w)\le \log \left| \frac{1-\overline{\varphi (0)}w}{\varphi (0)-w}%
\right| ,\hspace{.3cm}w\in \Delta \setminus \{\psi (0)\}
\label{littlewoodineq}
\end{equation}
with equality holding for quasi--every $w$ (i.e., all $w$ except those in a
set of capacity zero) exactly when $\psi $ is inner.
\end{theorem}
\noindent If $\psi (0)=0$, then (\ref{littlewoodineq}) reduces to
\begin{equation*}
N_\psi (w)\le -\log |w|
\end{equation*}
which is an improvement of the Schwarz inequality.
\bigskip
For counting functions on $\Omega $, we have the following generalization of
Littlewood's inequality:
\begin{theorem}
Let $\varphi :\Omega \rightarrow \Omega $ be analytic and fix the point $t_0$%
. Then
\begin{equation*}
N_\varphi (w)=\sum_{\varphi (z)=w}g(z;t_0)\leq g(w;t_0)\text{ for all }w\in
\Omega \backslash \left\{ t_0\right\} \text{,}
\end{equation*}
with equality holding (for quasi--every $w$) exactly when $\varphi (\Gamma
)\subset \Gamma $, by which we will mean that the boundary values of $%
\varphi $ on $\Gamma $ lie in $\Gamma $ almost everywhere (with respect to $%
\omega _{t_0}$).
\end{theorem}
\noindent \emph{Proof: }Let $g(z;w)$ be the Green's function for $\Omega $
with pole at $w$. The function $g(\varphi (z);w)$ is harmonic on $\Omega $
except at the collection of isolated points where $\varphi (z)=w$; at such a
point $g(\varphi (z);w)$ has a logarithmic pole. Let $^{*}\!g(\varphi (z);w)$
be the (multiple-valued) harmonic conjugate of $g(\varphi (z);w)$ on $\Omega
\setminus \{\varphi (z)=w\}$ and set
\begin{equation*}
Q_w(z)=e^{-g(\varphi (z);w)-i^{*}\!g(\varphi (z);w)}.
\end{equation*}
$Q_w$ lies in $MH^\infty $; indeed, its modulus is bounded by one. Using
Theorem \ref{factorization}, we factor $Q_w$ in $H^2(\Omega )$ as $%
Q_w(z)=B_w(z)S_w(z)F_w(z)$ where the factors are a Blaschke product, a
singular inner function, and an outer function, respectively. We then get
\begin{eqnarray*}
-\log \left| Q_w(t_0)\right| &=&g(\varphi (t_0);w) \\
&=&g(t_0;w) \\
&=&-\log \left| B_w(t_0)\right| -\log \left| S_w(t_0)\right| -\log \left|
F_w(t_0)\right| \text{.}
\end{eqnarray*}
The function $Q_w(z)$ has zeros exactly where $\varphi \left( z\right) =w$,
so we have
\begin{equation*}
-\log \left| B_w(t_0)\right| =\sum_{\varphi \left( z\right) =w}g\left(
t_0;z\right) =N_\varphi (w)\text{.}
\end{equation*}
Thus we see that
\begin{equation*}
g(w;t_0)=N_\varphi (w)-\log \left| S_w(t_0)\right| -\log \left|
F_w(t_0)\right| \text{,}
\end{equation*}
so
\begin{equation*}
g(w;t_0)\geq N_\varphi (w)\text{.}
\end{equation*}
We have equality when both $\log \left| S_w(t_0)\right| $ and $\log \left|
F_w(t_0)\right| $ are zero, which happens when both $S_w$ and $F_w$ are
unimodular constants. If $\varphi (\Gamma )\subset \Gamma $, then we will
have $|Q_w|=1$ almost everywhere on $\Gamma $ and thus $F_w\equiv 1$. By the
extension of Frostman's theorem which is proved below, Theorem \ref{frostgen}%
, since $Q_w$ is a Blaschke product composed with $\varphi $, it has trivial
singular factor for quasi--every $w$ in $\Omega $.\hfill$\square $
The well--known theorem of Frostman, for functions on the unit disk, can be
stated
\begin{theorem}
\label{frost}Let $\psi $ be an inner function on $\Delta $. Then for $\left|
w\right| <1$, the function
\begin{equation}
q_w(z)=\frac{\psi (z)-w}{1-\overline{w}\psi (z)} \label{blaschketerm}
\end{equation}
is a Blaschke product except possibly for a set of $w$ in $\Delta $ of
logarithmic capacity zero.
\end{theorem}
For our generalization, we will prove the following theorem and associated
lemma, which are suggested in \cite[Ch. 5, Exercise 2,3]{SDF}
\begin{theorem}
\label{frostgen}Let $\varphi $ be an analytic function on $\Omega $ with $%
\varphi \left( \Gamma \right) \subset \Gamma $, and $B_w(z)=\exp \left\{
-g(z;w)-i^{*}\!g(z;w)\right\} $ be the Blaschke product on $\Omega $ with
zero at $w$. Then the function
\begin{equation*}
Q_w(z)=B_w(\varphi (z))
\end{equation*}
is a Blaschke product (on $\Omega $), except possibly for a set of $w$ in $%
\Omega $ of logarithmic capacity zero.
\end{theorem}
\noindent\emph{Proof:} For $\Pi $ the universal covering map from $\Delta $
onto $\Omega $ (with $\Pi (0)=t_0$), we have the pull--back map $\psi
:\Delta \rightarrow \Delta $ which satisfies $\varphi \circ \Pi =\Pi \circ
\psi $. It is easy to see that if $\varphi \left( \Gamma \right) \subset
\Gamma $, then $\psi $ must be inner. Define
\begin{equation*}
E=\left\{ w\in \Delta :\frac{\psi (z)-w}{1-\overline{w}\psi (z)}\text{ has a
nontrivial singular factor}\right\} .
\end{equation*}
By Theorem \ref{frost} above, $E$ has logarithmic capacity zero, thus so
does $\Pi (E)$ (in $\Omega $). We write
\begin{eqnarray*}
Q_w\circ \Pi &=&B_w\circ \varphi \circ \Pi \\
&=&B_w\circ \Pi \circ \psi .
\end{eqnarray*}
By Lemma \ref{frostgenlemma}, below, $B_w\circ \Pi $ is a Blaschke product
on $\Delta $, with zeros at those points $z$ with $\Pi (z)=w$. The function $%
B_w\circ \Pi \circ \psi $ is thus a (constant times a) product of terms of
the form (\ref{blaschketerm}). If $w$ is not in $\Pi (E)$, then each of
these terms is a Blaschke product. Thus $B_w\circ \Pi \circ \psi =B_w\circ
\varphi \circ \Pi $ is a Blaschke product, and, again by Lemma \ref
{frostgenlemma}, $B_w\circ \varphi =Q_w$ is a Blaschke product on $\Omega $.%
\hfill$\square $
\begin{lemma}
\label{frostgenlemma}The analytic function $B$ on $\Omega $ is a Blaschke
product if and only if $B\circ \Pi $ is a Blaschke product on $\Delta $.
\end{lemma}
\noindent\emph{Proof:} If $B$ is a Blaschke product on $\Omega $, we can
write
\begin{equation*}
B(z)=e^{-\sum g(z;z_j)-i^{*}\!\left( \sum g(z;z_j)\right) }
\end{equation*}
for some sequence $\left\{ z_j\right\} $ with the property that $%
\sum_1^\infty g(\zeta ;z_j)<\infty $ for each $\zeta \in \Omega $. $B\circ
\Pi $ is easily seen to be an inner function on $\Delta $, so we can write
\begin{equation}
B\circ \Pi =bS, \label{blaschkefact}
\end{equation}
where $b$ is a Blaschke product on $\Delta $ and $S$ is a singular inner
function. The Blaschke product $b$ has a zero at any $z$ with $\Pi (z)=z_j$
for some $j$. We now see that
\begin{eqnarray*}
-\log \left| B\circ \Pi (0)\right| &=&-\log \left| B(t_0)\right| \\
&=&\sum_jg(t_0;z_j) \\
&=&\sum_j\sum_{\Pi (z)=z_j}\log \frac 1{\left| z\right| } \\
&=&-\log \left| b(0)\right| \text{.}
\end{eqnarray*}
The third line above comes from the fact \cite[VII.5]{RN} that we can write
the Green's function for $\Omega $ in terms of Green's functions on the unit
disk,
\begin{equation*}
g(w;t_0)=\sum_{\Pi (a)=w}\log \frac 1{\left| a\right| }.
\end{equation*}
But (\ref{blaschkefact}) gives us $-\log \left| B\circ \Pi (0)\right| =-\log
\left| b(0)\right| -\log \left| S(0)\right| $, so $\left| S(0)\right| =1$,
and thus $S\equiv 1$, i.e., $B\circ \Pi =b$ is a Blaschke product in $\Delta
$.
Now assume $B\circ \Pi $ is a Blaschke product on $\Delta $. It is easy to
see that $B$ must be an inner function on $\Omega $, so it has the
factorization in $H^\infty (\Omega )$,
\begin{equation*}
B=bS
\end{equation*}
where $b$ is a Blaschke product on $\Omega $ and $S$ is a singular inner
function on $\Omega $ (i.e., $S$ has boundary values of modulus $1$ a.e.,
and has no zeros on $\Omega $). We then have
\begin{equation*}
B\circ \Pi =\left( b\circ \Pi \right) \left( S\circ \Pi \right) ,
\end{equation*}
and we can easily see that $S\circ \Pi $ is a function on $\Delta $ which
has no zeros and has boundary values of $1$ a.e., so $S\circ \Pi $ is a
singular inner function. But $B\circ \Pi $ is a Blaschke product, so has
only trivial singular inner factor, i.e., $S\circ \Pi $ is trivial, so $S$
must be trivial, and $B$ must be a Blaschke product.\hfill$\square $
\subsection{The sub--mean--value property}
We will need the following property for the counting function on $\Omega $:
\begin{theorem}
\label{submeanval}Let $h$ be an analytic function on a domain $U$. Suppose
that $D$ is an open disk in $U\backslash h^{-1}(t_0)$ with center at $a$ and
that $h(D)\subset \Omega $. Then
\begin{equation}
N_\varphi (h(a))\leq \frac 1{A(D)}\int_DN_\varphi (h(w))dA(w)
\label{submeanvalprop}
\end{equation}
where $N_\varphi $ is the counting function for $\Omega $ and $A$ is area
measure.
\end{theorem}
\noindent\emph{Proof:} This sub--mean--value property follows from the
version proved in \cite{JHS}, since we can express our counting function on $%
\Omega $ as a counting function on the unit disk:
\begin{equation*}
N_\varphi (w)=\sum_{\varphi (z)=w}g(z;t_0).
\end{equation*}
As we did earlier, we write the Green's function of $\Omega $ in terms of
the Green's function on $\Delta $ to get
\begin{eqnarray*}
N_\varphi (w) &=&\sum_{\varphi (z)=w}g(z;t_0) \\
&=&\sum_{\varphi (z)=w}\sum_{\Pi (a)=z}\log \frac 1{\left| a\right| } \\
&=&\sum_{\varphi \circ \Pi (a)=w}\log \frac 1{\left| a\right| } \\
&=&N_{\varphi \circ \Pi }\left( w\right) ,
\end{eqnarray*}
for $\Pi $ the universal covering map of the unit disk onto $\Omega $ which
maps $0$ to $t_0$. In this last line, $N_{\varphi \circ \Pi }\left( w\right)
$ is the counting function on the unit disk. It is shown in \cite{JHS} that $%
N_{\varphi \circ \Pi }\left( h(w)\right) $ has the required sub--mean--value
property, so $N_\varphi \left( h(w)\right) $ has the same property.\hfill$%
\square $
\subsection{The Littlewood--Paley identity}
For functions in $H^2(\Delta )$, we have the Littlewood-Paley identity \cite
{JHS}:
\begin{theorem}
For functions $f\in H^2(\Delta )$,
\begin{equation*}
\left\| f\right\| _{H^2(\Delta )}^2=\frac 1{2\pi }\int_T\left| f(e^{i\theta
})\right| ^2d\theta =\frac 2\pi \int_\Delta \left| f^{\prime }(z)\right|
^2\log (1/\left| z\right| )dA(z)+\left| f(0)\right| ^2.
\end{equation*}
\end{theorem}
The corresponding theorem on $\Omega $ is
\begin{theorem}
\label{genlp}For functions $f\in H^2(\Omega )$,
\begin{equation*}
\left\| f\right\| _{H^2(\Omega )}^2=\int_\Gamma \left| f\right| ^2d\omega
_{t_0}=\frac 2\pi \int_\Omega \left| f^{\prime }(z)\right|
^2g(z;t_0)dA+\left| f(t_0)\right| ^2\text{,}
\end{equation*}
where $\omega _{t_0}$ is harmonic measure on $\Gamma $ for $t_0$.
\end{theorem}
\noindent\emph{Proof:} Let $r$ be a small positive number and let $\Omega
_r=\Omega \setminus \{z:|z-t_0|\le r\}$. The boundary of $\Omega _r$ is $%
\Gamma _r=\Gamma \cup \{z:|z-t_0|=r\}$. We begin with Green's formula:
\begin{equation*}
\int_{\Gamma _r}\left( u\frac{\partial v}{\partial n}-v\frac{\partial u}{%
\partial n}\right) ds=\int_{\Omega _r}\left( u\Delta v-v\Delta u\right) \,dA.
\end{equation*}
We take $u=|f|^2$ and $v=g(\cdot ;t_0)$. We have
\begin{equation*}
d\omega _{t_0}=\frac{-1}{2\pi }\frac{\partial v}{\partial n}\,ds\text{%
\hspace{.3cm} and \hspace{.3cm} }\Delta u=4|f^{\prime }|^2.
\end{equation*}
On the circle $|z-t_0|=r$, the normal derivative of $v$ is the radial
derivative and equals $1/r$ plus a bounded term. This gives a term on the
left--hand side of $2\pi |f(t_0)|^2$ as $r\to 0$. On the other hand, $v$
itself is $\log r$ plus a bounded term and so the other term from the
left--hand side goes to zero as $r\to 0$. On the right--hand side, the
Laplacian of $v$ is identically zero on $\Omega _r$ and $v$ itself is $\log
s $, $0~~=-=.
\end{equation*}
and so $u_0$ is orthogonal to $u_1$. Next, $u_2$ is orthogonal to $%
v_3,v_4,\dots $. We write $u_2=v_2+h_2$ where $h_2\in E_2$. Hence,
\begin{equation*}
0==-=
\end{equation*}
and
\begin{equation*}
0==-=
\end{equation*}
so that $u_2$ is orthogonal to both of $u_0$ and $u_1$. In a similar way we
can see that the functions $u_0,u_1,u_2,...$ are mutually orthogonal. Next,
it is easy to establish that the linear span of $u_0,...,u_n$ is the
orthogonal complement of $E_n,n=0,1,2,...$ and so if $v\in H^2$ is
orthogonal to $u_0,u_1,\dots $, then
\begin{equation*}
v\in \bigcap_{n=0}^\infty E_n.
\end{equation*}
and thus $v=0$. Finally, suppose $f\in H^2$ has an orthonormal expansion $%
f=\sum c_ku_k$. Those $u_k$ with $k>(p+1)m$ vanish at $t_0$ to order at
least $m$ and hence $f-\sum_0^{m(p+1)}c_ku_k$ has a zero at $t_0$ of order
at least $m$.
\bigskip
\begin{proposition}
\label{pointests}Suppose that $f\in H^2(\Omega )$ vanishes to order $n$ at $%
t_0$. Let $\Pi $ be the universal covering map from $\Delta $ to $\Omega $.
For $\zeta \in \Omega $ and $\Pi (z)=\zeta $, let $\lambda _\Omega $ be the
Poincar\'{e} metric for $\Omega $. Then
\end{proposition}
\begin{enumerate}
\item[(a)] \hspace{.1in} $|f(\zeta )|\le \frac{|z|^n}{\sqrt{1-|z|^2}}\left(
\lambda _\Omega (\zeta )||\Pi ||_\infty \right) ^{\frac 12}||f||_2$
\item[(b)] \hspace{.1in} $|f^{\prime }(\zeta )|\le \sqrt{2}n\frac{|z|^{n-1}%
}{\sqrt{1-|z|^2}}\left( \lambda _\Omega (\zeta )\right) ^{\frac
32}||f||_2||\Pi ||_\infty ^{\frac 12}{}.$
\end{enumerate}
\noindent \emph{Proof:} Let $g=f\circ \Pi $ so that $g$ has a zero of order $%
n$ at the origin. Then use the standard estimates (see \cite{JHS}) in the
unit disk plus the fact that $\lambda _\Omega (\zeta )=|\Pi ^{\prime
}(z)|(1-|z|^2)\le ||\Pi ||_{Bloch}\le ||\Pi ||_\infty $.
\section{The Main Theorem}
With the background of Section \ref{background} in place, we are now ready
to state and prove the main result of this paper. \bigskip
\begin{theorem}
\label{mainthm}Suppose that $\Omega $ is finitely connected and that $%
\varphi $ is an analytic function mapping $\Omega $ into itself with $%
\varphi (t_0)=t_0$. Let $\left\| C_\varphi \right\| _e$ denote the essential
norm of $C_\varphi $, regarded as an operator on $H^2\left( \Omega \right) $
. Then
\begin{equation*}
\left\| C_\varphi \right\| _e^2=\limsup_{w\rightarrow \Gamma }\frac{%
N_\varphi (w)}{g(w;t_0)}.
\end{equation*}
In particular, $C_\varphi $ is compact on $H^2$ if and only if
\begin{equation*}
\lim_{w\rightarrow \Gamma }\frac{N_\varphi (w)}{g(w;t_0)}=0.
\end{equation*}
\end{theorem}
\noindent We will prove the theorem by proving separately upper and lower
bounds for the essential norm of $C_\varphi $.
\subsection{The Upper Bound}
We will use the following general formula from \cite{JHS} for the essential
norm of a linear operator on a Hilbert space:
\begin{theorem}
\label{prop}Suppose $T$ is a bounded linear operator on a Hilbert space $H$.
Let $\left\{ K_n\right\} $ be a sequence of compact self--adjoint operators
on $H$, and write $R_n=I-K_n$. Suppose $\left\| R_n\right\| =1$ for each $n$%
, and $\left\| R_nx\right\| \rightarrow 0$ for each $x\in H$. Then $\left\|
T\right\| _e=\lim_n\left\| TR_n\right\| $.
\end{theorem}
The goal now is to show that, for an analytic function $\varphi :\Omega
\rightarrow \Omega $ which fixes the point $t_{0\text{,}}$
\begin{equation}
\left\| C_\varphi \right\| _e^2\leq \limsup_{w\rightarrow \Gamma }\frac{%
N_\varphi (w)}{g(w;t_0)}. \label{upperbound}
\end{equation}
We do this by applying the Proposition \ref{prop} above with $K_n$ the
operator which takes $f$ to the sum of the first $(p+1)n$ terms in its
expansion relative to the basis we have chosen for $H^2\left( \Omega \right)
$ in Theorem \ref{basis}.
For this orthonormal basis $u_0,u_1,u_2,\ldots $ of $H^2\left( \Omega
\right) $, we can write any $f\in H^2\left( \Omega \right) $ as $%
f=\sum_{k=0}^\infty c_ku_k$, and then $K_nf=\sum_{k=0}^{(p+1)n}c_ku_k$. $%
R_n=I-K_n$ will then be an operator with the property that $%
R_nf=\sum_{k=1+(p+1)n}^\infty c_ku_k$ has a zero of order at least $n$ at $%
t_{0\text{.}}$
The operator $K_n$ is self--adjoint and compact. Since $R_n=I-K_n$, its norm
is $1$, so that the hypotheses of the proposition are fulfilled, and
\begin{equation*}
\left\| C_\varphi \right\| _e=\lim_{n\rightarrow \infty }\left\| C_\varphi
R_n\right\| \text{.}
\end{equation*}
To estimate the right side of the above, fix a function $f$ in the unit ball
of $H^2\left( \Omega \right) $, and a positive integer $n$. Then by
Corollary \ref{cv} we get
\begin{equation*}
\left\| C_\varphi R_nf\right\| _{H^2(\Omega )}^2=\frac 2\pi \int \left|
\left( R_nf\right) ^{\prime }\right| ^2N_\varphi dA+\left| R_nf(\varphi
(t_0))\right| ^2.
\end{equation*}
Since $\left\| f\right\| _{H^2(\Omega )}\leq 1$, the same is true of $R_nf$.
Now fix $r<1$. Split the integral above into two parts, $\Omega _r=\Pi
(r\Delta )$ (where $\Pi $ is the universal covering map of $\Omega $ which
maps the origin to $t_0$), and the other its complement in $\Omega $, $%
\Omega _r^c$. Then take the supremum of both sides of the resulting
inequality over all functions $f$ in the unit ball $B$ of $H^2\left( \Omega
\right) $. We obtain
\begin{eqnarray*}
\left\| C_\varphi R_n\right\| ^2 &\leq &\sup_B\frac 2\pi \int_{\Omega
_r}\left| \left( R_nf\right) ^{\prime }\right| ^2N_\varphi dA \\
&&+\sup_B\frac 2\pi \int_{\Omega _r^c}\left| \left( R_nf\right) ^{\prime
}\right| ^2N_\varphi dA+\left| R_nf(\varphi (t_0))\right| ^2\text{.}
\end{eqnarray*}
We now use the pointwise estimate for $\left( R_nf\right) ^{\prime }$, from
Proposition \ref{pointests} part (b). $R_nf$ has a zero of order at least $n$
at $t_0$, and, for $w\in \Omega _r$, $w=\pi (z)$ for some $z$ with $\left|
z\right| ~~