\documentstyle[11pt]{article}
\title{Patch for\\Composition Operators\\on Spaces of Analytic Functions}
\author{Carl C. Cowen \mbox{ and } Barbara  D. MacCluer }
\addtolength{\textheight}{.12in}
\newcommand{\ph}{\mbox{$\varphi$}}
\newcommand{\ebox}{\hspace*{1em}\linebreak[0]\hspace*{1em}\hfill\rule[-1ex]{.1in}{.1in}\\}
\begin{document}
\maketitle

Exercise~7.5.2 (page 288) of our book {\em 
Composition Operators on Spaces of Analytic Functions}~\cite{book}
has an error: 
$\{e^{i\theta}: |\ph_n(e^{i\theta})|=1\}$  should be 
$\{\ph_n(e^{i\theta}): |\ph_n(e^{i\theta})|=1\}$.
The function $\ph(z)=z^3/(2-z^6)$ is an example of a function
that does not satisfy the conclusion of Exercise~7.5.2
as written: the fixed
points of \ph\ on the unit circle are $1$ and $-1$, but if 
$\zeta$ is a primitive
cube root of $1$, then $\ph_n(\zeta)=\ph_n(\zeta^2)=1$ and 
$\ph_n(-\zeta)=\ph_n(-\zeta^2)=-1$ for every $n$.

Because the result of this exercise is used in other places in the 
book, the error creates the necessity for changes in those places.
This patch fills the holes that the error leaves in the arguments.
A similar error occurs in~\cite[Theorem 4.7]{Ccomp} but {\em not\/}
in Kamowitz's paper~\cite{kamowitz75} where these results originated
(see also~\cite[page 289]{book}).  In the discussion that follows,
references by number are to results of~\cite{book}.

The first change to be made is that Theorem~2.64 needs to be
extended to cover points that are not necessarily fixed by 
\ph.\\[-1.5ex]

\noindent{\bf Corollary to Theorem 2.64}
{\em Suppose \ph, an analytic map of the 
unit disk into itself, has Denjoy--Wolff point $a$
with $|a|=1$ and $\ph'(a)<1$.  Suppose also that $b$ is another fixed point 
of \ph, that $b'$ is a point of the unit circle for which 
$\ph(b')=b$, that \ph\ is analytic in a neighborhood of $b$ 
and $b'$, and that there  is 
$\delta>0$ for which  $|\ph(z)|<1$ whenever $0<|z-b|<\delta$ and 
$|z|\leq 1$.  
If $\sigma$ is the map in the model that takes the disk into the right half 
plane and satisfies  $\sigma\circ\ph =\ph'(a)\sigma$, then for
every $p>|\log \ph'(a)/\log \ph'(b)|$, there is a constant  
$M'$ so that  
\[ |\sigma(z)|< M'|z-b'|^{-p} \]
in a neighborhood of $b'$.}\\

\noindent{\bf Proof.}
Since \ph\ is analytic in a neighborhood of $b'$ and 
$\ph'(b')\neq 0$, there is a neighborhood of $b'$ on which
\ph\ is a one--to--one map onto a neighborhood of $b$.  In
particular, by Theorem~2.64, for $z$ close enough to $b'$ in 
this neighborhood, $w=\ph(z)$ satisfies 
$|\sigma(w)|< M|w-b|^{-p}$.  Now 
\[ |\sigma(w)|=|\sigma(\ph(z))|=\ph'(a)|\sigma(z)| \]
and since \ph\ is differentiable at $b'$, for $z$ close enough
to $b'$,
\[ |w-b|=|\ph(z)-\ph(b')|\geq \frac{|\ph'(b')|}{2}|z-b'| \]
Thus, for $z$ close enough to $b'$, 
\[ |\sigma(z)|< M'|z-b'|^{-p} \]
where $M'=2^p\ph'(a)^{-1}|\ph'(b')|^{-p}M$.
\ebox

In the corollary above, notice that it is size of the derivative
at $b$ that is controlling the growth of $\sigma$, not the 
size of the derivative at $b'$!

We also need to revise Lemma~7.25 and the paragraph preceding
it.  

\begin{quote}
We are now ready to shrink the inner radius of the annulus of eigenvalues
for smooth functions.  By Exercise~7.5.1, if \ph\ is analytic in a 
neighborhood
of the closed unit disk and \ph\ is not inner, 
then $\{e^{i\theta}:|\ph(e^{i\theta})|=1\}$ is at most 
a finite set of points.  In the hypothesis of the next result
we assume that fixed points of \ph\ and their preimages are the 
only points of the circle that get mapped to the circle.
This eliminates inner functions, but, as we will see when we 
identify the spectrum, the result can be used to get information 
about smooth functions that are not inner.
\end{quote}

\noindent{\bf Lemma 7.25 (revised)\ }{\em 
Suppose  \ph\ is analytic in a neighborhood of the closed unit 
disk, maps the unit disk to itself, and has Denjoy--Wolff
point $a$ on the unit circle with $\ph'(a) < 1$.  
Suppose also that 
$\{e^{i\theta}: |\ph(e^{i\theta})| = 1\} = 
\{a, b_1,b_2,\dots,b_k,c_1,\dots,c_\ell\}$,
where $\ph(b_j)=b_j$ for $j=1,\dots,k$ and 
$\ph(\ph(c_j))=\ph(c_j)\neq c_j$ for $j=1,\dots,\ell$.  If
\[
\max \{\ph'(b_j)^{-1/2} : j = 1,2,\dots,k\} < |\lambda| < \ph'(a)^{-1/2}
\]
then $\lambda$ is an eigenvalue of $C_\varphi$ 
on $H^2(D)$ of infinite multiplicity.}

\noindent{\bf Proof.}  The $c_j$ in the hypothesis are not fixed
points but their images $\ph(c_j)$ are fixed points, either $a$
or one of the other fixed points. 

 The proof starting on line 6 of page 282
needs to include the $c_j$ in an appropriate way as well as the
fixed points.  For example, the display after line 8
should be
\[ \{\ph(z): |z|\leq 1 \mbox{ and } |z-b_j|\geq \epsilon 
\mbox{ and } |z-c_j|\geq \epsilon \mbox{ for all $j$}
 \} \]

In the argument that follows this display, we need to 
provide the same kind of growth estimates near the $c_j$
as have been provided near the $b_j$.  If $\ph(c_j)=a$, then
since $\lim_{w\rightarrow a}\sigma(w)=0$, we also have
$\lim_{z\rightarrow c_j}\sigma(z)=0$ and $\sigma^x$ is bounded
near $c_j$.  

The Corollary above shows that if  $b\neq a$ is a fixed 
point of \ph\ and $\ph(c_j)=b$, then there are $M_j'$ and
$\epsilon_j'$ so that 
\[ |\sigma(z)|\leq M_j'|z-c_j|^{-|\log s|/|2\log \lambda_0|}  \]
for $|z-c_j|<\epsilon_j'$. 

Now the product that provides the upper estimate for 
$\sigma^{2x}$ has terms for both the $b_j$'s and some of the 
$c_j$'s but is otherwise unchanged.  The conclusion follows as
before.
\ebox

The proof of Theorem~7.26, but not the statement, needs to 
be revised.  The sentence beginning on line 2 of the proof
should read
\begin{quote}
Furthermore, by Exercise~7.5.2 (revised), this implies that for 
some positive integer $n$,  the set 
$\{e^{i\theta} : |\ph_n(e^{i\theta})| = 1\}$ consists
entirely of fixed points $\{a, b_1,\dots,b_k\}$ of $\ph_n$
together with points $\{c_1,\dots,c_\ell\}$ that are not fixed
points of $\ph_n$ but $\ph_n(c_j)$ is a fixed point
of $\ph_n$.
\end{quote}
The remainder of the proof goes as before using the revised
version of Lemma~7.25.

Finally, the set ${\cal S}_n$ defined on page 296
needs to have its definition changed to 
\[ {\cal S}_n=\{ \ph_n(w) : |w|=1 \mbox{ and } |\ph_n(w)|=1\}   \]
It is this set that consists of fixed points of $\ph_n$ and
it is  this set that gives rise to the essential spectral radius of 
$C_\varphi$.

%
%**********************     REFERENCES  References       *****************
%
\begin{thebibliography}{99}
\bibitem[Co83]{Ccomp}{\sc C.~C.~Cowen,} {Composition operators on
$H^2$, {\em J. Operator Theory} {\bf 9}(1983), 77--106. }
\bibitem[CoM95]{book} {\sc C.~C.~Cowen {\rm and }B.~D.~MacCluer,} 
{{\em Composition
Operators on Spaces of Analytic Functions,} CRC Press, Boca Raton, 1995.}
\bibitem[Kam75]{kamowitz75}{\sc H.~Kamowitz,} {The spectra of composition operators 
on $H^p$, {\em J. Functional Analysis}
{\bf 18}(1975), 132--150.} 
\end{thebibliography}
\end{document}

