Geometry of the symmetrized polydisc
Armen Edigarian
We describe all proper holomorphic mappings of the symmetrized polydisc and study its geometric properties. We also apply the obtained results to the study of the spectral unit ball in $\MM_n(\CC^n)$.
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\begin{document}
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\title%[Proper holomorphic mappings]
{Geometry of the symmetrized polydisc}

\author{Armen Edigarian}

\address{Institute of Mathematics, Jagiellonian University,
Reymonta 4, 30-059 Kra\-k\'ow, Poland}
\email{Armen.Edigarian@im.uj.edu.pl}
\thanks{The authors were supported in part by the KBN grant
No. 5 P03A 033 21}

\author{W\l odzimierz Zwonek}

%\address{Institute of Mathematics, Jagiellonian University,
%Reymonta 4, 30-059 Krak\'ow, Poland}
\email{zwonek@im.uj.edu.pl}

%\date{26 December 2003}

%    General info
%%%\subjclass{Primary 54C40, 14E20; Secondary 46E25, 20C20}

%%%\dedicatory{This paper is dedicated to }

%%%\keywords{ }

\begin{abstract} We describe all proper holomorphic mappings of the
symmetrized polydisc and study its geometric properties. We also
apply the obtained results to the study of the spectral unit ball
in $\MM_n(\CC^n)$.
\end{abstract}

\maketitle

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\section{Introduction}
Let $\DD$ be the unit disc in the complex plane $\CC$. Let
$\pi_n=(\pi_{n,1},\dots,\pi_{n,n}):\CC^n\to\CC^n$, $n\ge1$, be
defined as follows
\begin{equation*}
\pi_{n,k}(\lambda_1,\dots,\lambda_n)= \sum_{1\le j_1<\dots<j_k\le
n}\lambda_{j_1}\dots\lambda_{j_k},\quad k=1,\dots,n.
\end{equation*}



Observe that $\pi_n$ is a proper holomorphic mapping
with the multiplicity equal to $n!$ (see e.g. \cite{Rud}).
Moreover, $\pi_n^{-1}\big(\pi_n(\DD^n)\big)=\DD^n$.
Hence, $\pi_n|_{\DD^n}:\DD^n\to\pi_n(\DD^n)$ is a proper
holomorphic mapping. Put $\GG_n=\pi_n(\DD^n)$ and
$\delta_n=\pi_n\big((\partial\DD)^n\big)$. The domain $\GG_n$ is
called the {\it symmetrized $n$-disc}.

Below we present a number of
results on the geometry of the symmetrized polydisc $\GG_n$.

One of the main results of the paper is to give the following
characterization of proper holomorphic self-mappings of the
symmetrized polydisc.
\begin{theorem}\label{thm:1} Let $f:\GG_n\to\GG_n$ be a holomorphic mapping. Then
$f$ is proper if and only if there exists a finite Blaschke
product $B$ such that
\begin{equation*}
f\big(\pi_n(\lambda_1,\dots,\lambda_n)\big)=
\pi_n\big(B(\lambda_1),\dots,B(\lambda_n)\big),\quad
\lambda_1,\dots,\lambda_n\in\DD.
\end{equation*}
In particular, $f$ is an automorphism if and only if
\begin{equation*}
f\big(\pi_n(\lambda_1,\dots,\lambda_n)\big)=
\pi_n(h(\lambda_1),\dots,h(\lambda_n)\big), \quad
\lambda_1,\dots,\lambda_n\in\DD,
\end{equation*}
where $h$ is an automorphism of the unit disc $\DD$.
\end{theorem}
Note that, if $\psi:\DD\to\DD$ is a holomorphic function, then the
mapping $f_\psi:\GG_n\to\GG_n$, defined as
\begin{equation*}
f_\psi\big(\pi_n(\lambda_1,\dots,\lambda_n)\big)=
\pi_n\big(\psi(\lambda_1),\dots,\psi(\lambda_n)\big),
\quad\lambda_1,\dots,\lambda_n\in\DD,
\end{equation*}
is a well-defined holomorphic mapping. Moreover, $f_\psi$ is
proper (resp.~an automorphism) if and only if $\psi$ is proper
(resp.~an automorphism).

We get Theorem~\ref{thm:1} as a corollary of the following
\begin{theorem}\label{thm:2} Let $f:\DD^n\to\GG_n$ be a holomorphic mapping.
Then $f$ is proper if and only if there exist finite Blaschke
products $B_1$, \dots, $B_n$ such that
\begin{equation*}
f(\lambda_1,\dots,\lambda_n)=\pi_n\big(
B_1(\lambda_1),\dots,B_n(\lambda_n)\big), \quad
\lambda_1,\dots,\lambda_n\in\DD.
\end{equation*}
\end{theorem}

The symmetrized polydisc has been recently studied by many
authors, especially in two-dimensional case (see
e.g.~\cite{Agl-You1, Agl-You2, Agl-You3}, \cite{Cos1},
\cite{Pfl-Zwo}). Description of automorphisms in $\GG_2$ was given
in \cite{Jar-Pfl2}, description of proper mappings in $\GG_2$ was
given in \cite{Edi}. We follow the ideas of the latter paper.

% Note that the singular set is equal to
%$\Sigma=\{\pi(\lambda,\lambda,\mu):\mu,\lambda\in\CC\}$.
%For the set $K\subset I_n:=\{1,\ldots,n\}$ define $\sum_{K}$ to be
%the set of permutations $\sigma$ of the set $I_n$ ($\sigma\in
%S_n$) such that $\sigma(K)=K$. For $\sigma\in S_n$,
%$\lambda=(\lambda_1,\ldots,\lambda_n)\in\CC^n$ define
%$\lambda_{\sigma}:=(\lambda_{\sigma(1)},\ldots,\lambda_{\sigma(n)})$.

\section{Proofs}

For the set $K=\{j_1,\ldots,j_k\}\subset I_n:=\{1,\ldots,n\}$,
$1\leq j_1<\ldots<j_k\leq n$ and
$\lambda=(\lambda_1,\ldots,\lambda_n)\in\CC^n$ define
$\lambda_K:=(\lambda_{j_1},\ldots,\lambda_{j_k})$.

Let $\lambda_0\in\CC$, $0\le k\le n$. We define $D_k^n(\lambda_0)$
to be the domain in $\CC^n$ of all
$\lambda=(\lambda_1,\ldots,\lambda_n)\in\CC^n$ such that there is
a (uniquely defined) set $K=K(\lambda_0)\subset I_n$ with $\#K=k$
and $|\lambda_j-\lambda_0|<|\lambda_k-\lambda_0|$, $j\in K$, $k\in
I_n\setminus K$.

Note that $D_n^n(\lambda_0)=D_0^n(\lambda_0)=\CC^n$ for any
$\lambda_0\in\CC$.

\begin{remark}\label{rem:3} Let us fix $\lambda_0\in\CC$, $0\leq k\leq n$. It is
immediate to see that the mappings
\begin{multline}
\rho_{k,n}:D_k^n(\lambda_0)\owns\lambda\mapsto\pi_k(\lambda_K)\in\CC^k,\\
\tilde\rho_{k,n}:D_k^n(\lambda_0)\owns\lambda\mapsto\pi_{n-k}(\lambda_{I_n\setminus
K})\in\CC^{n-k}
\end{multline}
are holomorphic. Moreover, for any mapping
$\phi\in\OO(D,\pi_n(D_k^n(\lambda_0)))$, where $D$ is a domain in
$\CC^m$, the mappings $\rho_{k,n}\circ\pi_n^{-1}\circ\phi$ and
$\tilde\rho_{k,n}\circ\pi_n^{-1}\circ\phi$ are holomorphic, too.
\end{remark}

\begin{lemma}\label{lem:4} Let $\phi\in\OO(D,\CC^n)$, where $D$ is a domain
in $\CC^m$. Assume that $\phi(D)\subset\delta_n$. Then $\phi$ is
constant.
\end{lemma}
\begin{proof} We use induction on $n$. The case $n=1$ is trivial.
So assume that $n\geq 2$ and that lemma is valid for dimensions
$1,2,\ldots,n-1$. Let $f=(f_1,\ldots,f_n)$ denote the multi-valued
mapping $\pi_n^{-1}\circ\phi$. Let $k$ denote the maximal number
of elements of a set $K\subset I_n$ such that for some
$\mu_0\in\DD$ all the coordinates $f_j(\mu_0)$, $j\in K$, are
equal. Without loss of generality we may assume that
$f_1(\mu_0)=\ldots=f_k(\mu_0)$ and $f_l(\mu_0)\neq f_1(\mu_0)$,
$l\geq k+1$. Shrinking $D$, if necessary, we may assume that
$\phi(D)\subset\pi_{n}(D_k^n(f_1(\mu_0)))$, so in view of
Remark~\ref{rem:3} (with $\lambda_0=f_1(\mu_0)$) we get that
$\rho_{k,n}\circ f$ is holomorphic. In particular,
$h:=(\rho_{k,n}\circ f)_1$ is holomorphic on $D$ and $|h|$ attains
its maximum (equal to $k$) at $0$. Therefore, losing no
generality, we may assume that $f_1,\ldots,f_k$ are constant on
$D$. In the case $k=n$ this finishes the proof. So assume that
$1\leq k<n$. Note that the function
$\tilde\phi:=\tilde\rho_{k,n}\circ f$ is holomorphic on $D$ and
$\tilde\phi(D)\subset\delta_{n-k}$. Then the inductive assumption
implies that $\tilde\phi$ is constant, which easily implies that
$f_{k+1},\ldots,f_n$ are constant, which finishes the proof.
\end{proof}

Let us define one more mapping. For $0\leq k\leq n$ and for
$w=\pi_k(\lambda)$, $z=\pi_{n-k}(\mu)$ define
\begin{equation}
p_{k,n}:\CC^k\times\CC^{n-k}\owns(w,z)\mapsto\pi_n(\lambda,\mu)\in\CC^n.
\end{equation}
Note that $p_{k,n}$ is a holomorphic mapping,
$p_{k,n}(\GG_k\times\GG_{n-k})=\GG_n$, $p_{k,n}(\bar
\GG_k\times\bar\GG_{n-k})=\bar\GG_n$.

\begin{lemma}\label{lem:5}
Let $\phi\in\OO(D,\CC^n)$, where $D$ is a domain in $\CC^m$, be such that
$\phi(D)\subset\partial\GG_n$. Then there are a $k$ with $1\leq
k\leq n$, $\theta\in\delta_k$, and $\psi\in\OO(D,\GG_{n-k})$ such
that $\phi=p_{k,n}\circ(\theta,\psi)$.
\end{lemma}
\begin{proof} The case $n=1$ is trivial. So assume that $n\geq 2$.
Define as in Lemma~\ref{lem:4}
$(f_1,\ldots,f_n):=\pi_n^{-1}\circ\phi$. Put $N(\lambda):=\#\{j\in
I_n:|f_j(\lambda)|=1\}$, $\lambda\in D$. Define
$k:=\max\{N(\lambda):\lambda\in D\}$. Let $k=N(\mu_0)$, $\mu_0\in
D$. Obviously, $1\leq k\leq n$.

Denote $u(\lambda):=\max\sb{1\leq j_1<\ldots<j_k\leq
n}\{|f_{j_1}(\lambda)|\cdot\ldots \cdot|f_{j_k}(\lambda)|\}$,
$\lambda\in D$. Then $u$ is plurisubharmonic in $D$, $u\leq 1$ on
$D$ and $u(\mu_0)=1$. Then $u\equiv 1$ on $D$. Therefore, for any
$\lambda\in D$ there is a set $K\subset I_n$ with $k$ elements
such that $|f_j(\lambda)|=1$, $j\in K$ and $|f_j(\lambda)|<1$,
$j\in I_n\setminus K$. Then Lemma~\ref{lem:4} finishes the proof
in the case $k=n$ (because $\phi(D)\subset\delta_n$). So assume
that $k<n$. Applying Remark~\ref{rem:3} (for $\lambda_0=0$) we see
that $\phi=p_{k,n}\circ(\eta,\psi)$, where $\eta=\rho_{k,n}\circ
f$, $\psi=\tilde\rho_{k,n}\circ f$ are holomorphic on $D$. It
easily follows from the definition that $\psi(D)\subset\GG_{n-k}$.
Moreover, $\eta(D)\subset\delta_k$, so, in view of
Lemma~\ref{lem:4}, $\eta$ is constant.
\end{proof}


Let $\lambda=(\lambda_1,\dots,\lambda_n)\in\CC^n$. Note that
$\lambda_1,\dots,\lambda_n$ are roots of the polynomial equation
\begin{multline}
z^n-\big(\pi_n(\lambda)\big)_1z^{n-1}+ \big(\pi_n(\lambda)\big)_2
z^{n-2}+\dots+(-1)^{n-1}\big(\pi_n(\lambda)\big)_{n-1}z\\
+(-1)^{n}\big(\pi_n(\lambda)\big)_n=0,\quad z\in\CC.
\end{multline}

Therefore, Lemma~\ref{lem:5} easily implies the following
\begin{lemma}\label{lem:6} Let $\phi\in\OO(D,\CC^n)$, where $D$ is a domain in
$\CC^m$, be such that $\phi(D)\subset\partial\GG_n$. Then there is
a constant $C\in\partial\DD$ such that
\begin{multline}
C^n-\phi_1(\lambda)C^{n-1}+\phi_2(\lambda)C^{n-2}+\dots\\
+(-1)^{n-1}\phi_{n-1}(\lambda)C+ (-1)^{n}\phi_n(\lambda)=0,\quad
\lambda\in D.
\end{multline}
\end{lemma}

The proof of Theorem~\ref{thm:2} is based on the following result.
\begin{prop}\label{prop:7} Let $f:\DD^n\to\GG_n$ be a proper holomorphic
mapping. Then there exists a bounded holomorphic function $B$ on
$\DD$ such that
\begin{multline}
B^n(z_n)-B^{n-1}(z_n)f_1(z)+\dots+(-1)^{n-1}B(z_n)f_{n-1}(z)+(-1)^nf_n(z)
=0\\
z=(z_1,\dots,z_n)\in \DD^n.
\end{multline}
Moreover, $B$ is non-constant.
\end{prop}

\begin{proof} We use methods similar to the ones of Remmert-Stein
(see e.g. \cite{N}). Take a sequence $\DD\ni
z_n^{\nu}\to\partial\DD$. Then there exists a subsequence $\nu_k$
such that $f(\cdot,z_n^{\nu_k})\to\phi$ locally uniformly on
$\DD^{n-1}$. Note that $\phi:\DD^{n-1}\to\partial\GG_n$ is a
holomorphic mapping. Hence, in view of Lemma~\ref{lem:6} there
exists a constant $C\in\partial\DD$ such that
\begin{equation}\label{eq:1}
C^n-C^{n-1}\phi_1+\dots+(-1)^{n-1}C\phi_{n-1}+(-1)^n\phi_n\equiv0\quad\text{
on } \DD^{n-1}.
\end{equation}

Note that
\begin{multline}\label{eq:2}
-C^{n-1}\frac{\partial \phi_1}{\partial
z_j}+\dots+(-1)^{n-1}C\frac{\partial \phi_{n-1}}{\partial
z_j}+(-1)^n\frac{\partial \phi_n}{\partial z_j}\equiv0 \quad\text{
on } \DD^{n-1}\\ j=1,2,\dots,n-1.
\end{multline}

Since $f$ is proper, we have
\begin{equation*}
\det\left[\dif{f_i}{z_j}\right]_{i,j=1,\dots,n}\not\equiv0\quad\text{
on }\DD^n.
\end{equation*}
So, there exists a $k\in\{1,\dots,n\}$ such that
\begin{equation*}
\det\left[\dif{f_i}{z_j}\right]_{i=1,\dots,n, j=1,\dots,n-1, i\ne
k}\not\equiv0\quad\text{ on }\DD^n.
\end{equation*}
We want to solve the equations (w.r.t. $C$)
\begin{equation}
\begin{cases}
\sum_{m=1,m\ne k}^n (-1)^m C^{n-m}\dif{\phi_m}{z_1}=(-1)^{k+1}C^{n-k}\dif{\phi_k}{z_1}\\
\dots \dots\\
\sum_{m=1,m\ne k}^n (-1)^m
C^{n-m}\dif{\phi_m}{z_{n-1}}=(-1)^{k+1}C^{n-k}\dif{\phi_k}{z_{n-1}}
\end{cases}.
\end{equation}

Let
\begin{equation*}
\Phi_m(\phi_1,\dots,\phi_n):=
\det\left[\dif{\phi_i}{z_j}\right]_{i=1,\dots,n, i\ne m,
j=1,\dots,n-1}\text{ on $\DD^{n-1}$}.
\end{equation*}
Then $(-1)^m C^{n-m}\Phi_k(\phi_1,\dots,\phi_n)=(-1)^k
C^{n-k}\Phi_m(\phi_1,\dots,\phi_n)$.

If $k>1$ then take $m=k-1$. Hence,
\begin{equation}\label{eq:3}
\Phi_{k-1}(\phi_1,\dots,\phi_n)=-C\Phi_k(\phi_1,\dots,\phi_n)\quad
\text{ on }\DD^{n-1}.
\end{equation}
We have
\begin{multline}\label{eq:17}
\Phi_{k-1}(\phi_1,\dots,\phi_n)
\dif{\Phi_{k}(\phi_1,\dots,\phi_n)}{z_j}=\\
\Phi_{k}(\phi_1,\dots,\phi_n)\dif{\Phi_{k-1}(\phi_1,\dots,\phi_n)}{z_j}
\quad\text{ on }\DD^{n-1}
\end{multline}
for any $j=1,\dots,n-1$. Note that the equations in \eqref{eq:17}
hold for any choice of possible sequences $z_n^\nu\to\partial\DD$.
Therefore, the maximum principle for holomorphic functions implies
that similar equations hold for $f_1,\dots,f_n$. So,
\begin{equation}
\Phi_{k-1}(f_1,\dots,f_n) \dif{\Phi_{k}(f_1,\dots,f_n)}{z_j}=
\Phi_{k}(f_1,\dots,f_n)\dif{\Phi_{k-1}(f_1,\dots,f_n)}{z_j} \text{
on $\DD^n$}
\end{equation}
for any $j=1,\dots,n-1$. Put $A=\{\Phi_k(f_1,\dots,f_n)=0\}$ and
\begin{equation}
B=-\frac{\Phi_{k-1}(f_1,\dots,f_n)}{\Phi_{k}(f_1,\dots,f_n)}.
\end{equation}
Note that $A$ is a proper analytic subset of $\DD^n$ and that $B$
is a holomorphic function on $\DD^n\setminus A$. Moreover,
$\dif{B}{z_j}=0$ on $\DD^n\setminus A$ for any $j=1,\dots,n-1$.
Hence, $B$ depends only on $z_n$.



% and a
%holomorphic function $B\in\OO(\DD^3\setminus A)$ such that
%\begin{equation}\label{eq:4}
%\Phi_k(f_1,\dots,f_n)=-B(z_n)\Phi_{k+1}(f_1,\dots,f_n)\quad\text{
%on }\DD^3\setminus A.
%\end{equation}
Moreover, (use \eqref{eq:1}, \eqref{eq:3})
\begin{equation*}
\Phi_{k-1}^n+\phi_1\Phi_{k-1}^{n-1}\Phi_{k}+\dots+\phi_{n-1}\Phi_{k-1}\Phi_{k}^{n-1}+
\phi_n\Phi_{k}^n\equiv0\quad\text{ on }\DD^{n-1}.
\end{equation*}
Hence, similar result holds for $f_1,\dots,f_n$. From this we get
\begin{equation}
B^n(z_n)-B^{n-1}(z_n)f_1+\dots+(-1)^{n-1}B(z_n)f_{n-1}+(-1)^nf_n\equiv0\quad\text{
on } \DD^n\setminus A.
\end{equation}
Note that $B$ is a bounded function on $\DD^n\setminus A$, so it
extends holomorphically to $\DD^n$.

If $k=1$ then take $m=2$. Later we prove in a similar way.
\end{proof}


\begin{proof}[Proof of Theorem~\ref{thm:1}]
From Proposition~\ref{prop:7} we get that there are holomorphic
functions $B_1,\dots,B_n$ defined on $\Bbb D$ such that
\begin{multline}
B_m^n(z_m)-B_m^{n-1}(z_m)f_1(z)+\dots+(-1)^{n-1}B_m(z_m)f_{n-1}(z)+
(-1)^nf_n(z)\equiv0\\
z=(z_1,\dots,z_m)\in\DD^n,\ m=1,\dots,n.
\end{multline}
Hence, $f=\pi\big(B_1,\dots,B_n)$. So,
$B=(B_1,\dots,B_n):\DD^n\to\DD^n$ is a proper holomorphic mapping.
From this we have the proof.
\end{proof}



\section{The Shilov boundary}

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%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


We start with the description of the Shilov boundary of $\GG_n$.
The description of the Shilov boundary in the special case $n=2$
can be found in \cite{Jar-Pfl4} (see also \cite{Agl-You2}).

\begin{lemma} The set $\delta_n$ is the Shilov boundary of
$\GG_n$.
\end{lemma}
\begin{proof} It is easy to see that the modulus of any function from
$\cC(\bar\GG_n)\cap\OO(\GG_n)$ attains its maximum in
$\delta_n$. To finish the proof it is sufficient to show that for
any $z^0\in\delta_n$ there is a function
$F\in\cC(\bar\GG_n)\cap\OO(\GG_n)$ such that $|F|$ attains its
strict maximum at $z^0$.

Fix $z^0:=\pi_n(\lambda_1^0,\ldots,\lambda_n^0)\in\delta_n$,
$|\lambda_1^0|=\ldots=|\lambda_n^0|=1$. Let $B_1$ be a finite
Blaschke product such that
$B_1(\lambda_1^0)=\ldots=B_1(\lambda_n^0)=1$ (see \cite{Abr-Fis},
\cite{You}). Let $B_1^{-1}(1)=\{\lambda_1^0,\ldots,\lambda_m^0\}$,
where $m\geq n$. Let $B_2$ be a finite Blaschke product such that
$B_2(\lambda_1^0)=\ldots=B_2(\lambda_n^0)=1$ and
$B_2(\lambda_{n+1}^0)=\ldots=B_2^0(\lambda_{m})=-1$ (use once more
\cite{Abr-Fis}, \cite{You}). Let $B:=B_1+B_2$. Note that $|B|\leq
2$ on $\bar\DD$ and
$B^{-1}(2)=\{\lambda_1^0,\ldots,\lambda_n^0\}$. Define
$$F(z):=1+\sum\sb{j=1}\sp{n}B(\lambda_j),
$$
where $z=\pi(\lambda_1,\ldots,\lambda_n)\in\bar\GG_n$,
$\lambda_1,\ldots,\lambda_n\in\bar\DD$. Then
$F\in\cC(\bar\GG_n)\cap\OO(\GG_n)$ and $|F|$ attains its strict
maximum (equal to $1+2n$) at $z^0$.
\end{proof}


\section{The Bergman kernel}

It easily follows from the properties of proper holomorphic
mappings that
\begin{multline} \mathcal J_n:=\{\lambda\in\CC^n:\det
\pi^{\prime}(\lambda)=0\}=\\
\{\lambda\in\CC^n:\lambda_j=\lambda_k\text{ for some $j\neq k$,
$j,k=1,\ldots,n$.}\}.
\end{multline}
Denote by $K_D$ the Bergman kernel of the domain $D\subset\CC^n$
(see e.g. \cite{Jar-Pfl3}).
\begin{prop}\label{prop:9}
$$
K_{\GG_n}(\pi_n(\lambda),\pi_{n}(\mu))=
\frac{\det\left[\frac{1}{(1-\lambda_j\bar\mu_k)^2}\right]_{1\leq
j,k\leq
n}}{\pi^n\det\pi_n^{\prime}(\lambda)\det\overline{\pi_n^{\prime}(\mu)}}
$$
for any $\lambda,\mu\in\DD_n\setminus\mathcal J_n$.
\end{prop}

Let $\Sigma_n$ denote the group of all permutations of the set
$I_n$. For $\sigma\in\Sigma_n$,
$\lambda=(\lambda_1,\ldots,\lambda_n)\in\CC^n$ denote
$\lambda_{\sigma}:=(\lambda_{\sigma(1)},\ldots,\lambda_{\sigma(n)})$.

\begin{proof} From the formula for the Bergman kernel of the
polydisc and from the formula for the behavior of the Bergman
kernel under proper holomorphic mappings (see \cite{Bell}) we get
for any $\lambda,\mu\in\DD_n\setminus\mathcal J_n$
\begin{multline} K_{\GG_n}(\pi_n(\lambda),\pi_{n}(\mu))=
\frac{1}{\det\pi_n^{\prime}(\lambda)}\sum\sb{\sigma\in\Sigma_n}
K_{\DD^n}(\lambda,\mu_{\sigma})\frac{1}{\det\overline{\pi_n^{\prime}(\mu_{\sigma})}}
=\\
\frac{1}{\pi^n\det\pi_n^{\prime}(\lambda)\det\overline{\pi_n^{\prime}(\mu)}}
\sum\sb{\sigma\in\Sigma_n}\frac{(-1)^{\sgn\sigma}}{(1-\lambda_j\bar\mu_{\sigma(k)})^2}=
\frac{\det\left[\frac{1}{(1-\lambda_j\bar\mu_k)^2}\right]_{1\leq
j,k\leq
n}}{\pi^n\det\pi_n^{\prime}(\lambda)\det\overline{\pi_n^{\prime}(\mu)}}.
\end{multline}
\end{proof}

The formula above extends analytically to a formula on
$\GG_n\times\GG_n$. It would be interesting to find a more handy
formula for $K_{\GG_n}$. Below we deliver such a formula in the
case $n=2$. We start with the simplification of the denominator in
the formula for $K_{\GG_n}$.

\begin{lemma}\label{lem:10}
$\det\pi_1'(\lambda)=1$, $\lambda\in\CC$, and
$\det\pi_n^{\prime}(\lambda)=\prod_{1\leq j<k\leq
n}(\lambda_j-\lambda_k)$,
$\lambda=(\lambda_1,\dots,\lambda_n)\in\CC^n$, for any $n\ge2$.
\end{lemma}

\begin{proof} We prove by induction on $n$. For $n=2$ we get it by easy computations.
Put $\pi_{n,0}\equiv1$ for any $n\ge1$. Note that
\begin{equation}
\dif{\pi_{n,k}}{\lambda_1}(\lambda_1,\dots,\lambda_n)-
\dif{\pi_{n,k}}{\lambda_n}(\lambda_1,\dots,\lambda_n)=
(\lambda_n-\lambda_1)
\dif{\pi_{n-1,k-1}}{\lambda_1}(\lambda_1,\dots,\lambda_{n-1}).
\end{equation}
Similar equation holds for any pair $(\lambda_j,\lambda_n)$. From
this we get
\begin{multline}
\det\pi_n^{\prime}(\lambda_1,\dots,\lambda_n)= \det\Big[
\dif{\pi_{n,k}}{\lambda_j}(\lambda_1,\dots,\lambda_n)\Big]_{j,k=1,\dots,n}\\
=(\lambda_1-\lambda_n)\dots(\lambda_{n-1}-\lambda_n) \det\Big[
\dif{\pi_{n-1,k}}{\lambda_j}(\lambda_1,\dots,\lambda_{n-1})\Big]_{j,k=1,\dots,n-1}\\
= (\lambda_1-\lambda_n)\dots(\lambda_{n-1}-\lambda_n)
\det\pi_{n-1}^{\prime}(\lambda_1,\dots,\lambda_{n-1}).
\end{multline}
\end{proof}

%Consider
%$\tilde\lambda=(\lambda_1,\ldots,\lambda_{n-1})\in\CC^{n-1}\setminus\mathcal
%J_{n-1}$. Define
%\begin{equation*}
%p(\lambda_n):=\det\pi_n(\tilde\lambda,\lambda_n).
%\end{equation*}
%Then $p$ is a
%polynomial of degree not larger than $n-1$. Moreover,
%$p(\lambda_1)=\ldots=p(\lambda_{n-1})=0$. Consequently,
%$p(\lambda_n)=C_n(\tilde\lambda)\prod_{j=1}^{n-1}(\lambda_n-\lambda_j)$,
%$\lambda_n\in\CC$, where $C:\CC^{n-1}\setminus\mathcal
%J_{n-1}\mapsto\CC$ is holomorphic. This implies that the
%holomorphic function $w$ defined by the equality
%$w(\lambda):=\frac{\det\pi_n^{\prime}(\lambda)}{\prod\sb{1\leq
%j<k\leq n}(\lambda_k-\lambda_j)}$,
%$\lambda=(\lambda_1,\ldots,\lambda_n)\in\CC^n\setminus\mathcal
%J_n$, satisfies the equality $$\frac{\partial
%w(\lambda)}{\partial\lambda_n}=0,\;
%\lambda\in\CC^n\setminus\mathcal J_n.$$ Analoguously, we get the
%equality
%$$\frac{\partial
%w(\lambda)}{\partial\lambda_j}=0,\; \lambda\in\CC^n\setminus\mathcal J_n,\;j=1,\ldots,n.$$
%
%Consequently, $w$ is constant. So
%$\det\pi^{\prime}(\lambda)=C\prod\sb{1\leq j<k\leq
%n}(\lambda_k-\lambda_j)$, $\lambda\in\CC^n$.
%
%
%?????????????Now it is easy to verify that
%$\det\pi^{\prime}(\epsilon)=V(\{\epsilon_0,\ldots,\epsilon_{n-1})=1$,
%where $\epsilon=(\epsilon_0,\ldots,\epsilon_{n-1})$,
%$\root{n}\of{1}=\{\epsilon_0,\ldots,\epsilon_{n-1}\}$.
%???????????????
%\end{proof}

In the case $n=2$ elementary calculation shows.
\begin{prop}
\begin{multline}\label{eq:25}
K_{\GG_2}(\pi_2(\lambda),\pi_2(\mu))=\frac{2-(\bar\mu_1+\bar\mu_2)(\lambda_1+\lambda_2)+
2\lambda_1\lambda_2\bar\mu_1\bar\mu_2}{\pi^2((1-\lambda_1\bar\mu_1)(1-\lambda_2\bar\mu_2)(1-\lambda_1\bar\mu_2)
(1-\lambda_2\bar\mu_2))^2},\\
\lambda,\mu\in\DD_2.
\end{multline}
In particular, $\GG_2$ is the Lu-Qi-Keng domain, i.e.
$K_{\GG_2}(z,w)\neq 0$ for any $z,w\in\GG_2$.
\end{prop}

\begin{proof} To get the desired formula it is sufficient to apply
the formula from Proposition~\ref{prop:9}, Lemma~\ref{lem:10}, and
then make elementary calculations. To prove that the domain
$\GG_2$ is Lu Qi-keng, it is sufficient, in view of the form of
the automorphisms of $\GG_2$, to verify that
$K_{\GG_2}(\pi_2(\lambda),\pi_2(\mu))\neq 0$,
$\lambda,\mu\in\DD^2$ under the additional assumption $\mu_2=0$.
But this easily follows from the obtained formula \eqref{eq:25}.
\end{proof}

\begin{remark} It would be interesting to find a more
effective formula for the Bergman kernel for $\GG_n$, $n\geq
3$. Moreover, the problem whether the domain $\GG_n$, $n\geq
3$, is Lu Qi-keng, is open, too.
\end{remark}


\section{The spectral ball}





Define $\Omega_n:=\{W\in\mathcal M_n(\CC):r(W)<1\}$, where $r(W)$
denotes the spectral radius in $\mathcal M_n(\CC)$.

Denote also the following mapping
$$
\Psi_n:\Omega_n\owns W\mapsto\pi_n(\sigma(W))\in\GG_n,
$$
where $\sigma(W)$ denotes the spectrum of $W$. Note that $\Psi_n$
is a holomorphic mapping, which is onto but not one-to-one.


\begin{lemma}\label{lem:13} Let $W\in\Omega_n$. Put
$\sigma(W):=\{\lambda_1,\ldots,\lambda_n\}$,
$\lambda_1,\ldots,\lambda_n\in\CC$ (it may happen that
$\lambda_j=\lambda_k$ for some $j\neq k$). Then there is a
holomorphic mapping $f:\CC\mapsto \mathcal M_n(\CC)$ such that
$f(0)=W$, $f(1)=\diag(\lambda_1,\ldots,\lambda_n)$ and
$\sigma(f(\lambda))=\{\lambda_1,\ldots,\lambda_n\}$ for any
$\lambda\in\CC$.
\end{lemma}

\begin{proof} We proceed as in \cite{Cos}. There is an invertible
matrix $X\in\mathcal M_n(\CC)$ and an upper triangular matrix
$$
S=\bmatrix\lambda_1 & x_{1,2} & x_{1,3} & \ldots & x_{1,n-1} & x_{1,n}\\
          0       &\lambda_2 & x_{2,3} & \ldots & x_{2,n-1} & x_{2,n}\\
          \ldots & \ldots & \ldots & \ldots & \ldots & \ldots\\
          0      & 0 & 0 & \ldots &\lambda_{n-1} & x_{n-1,n}\\
          0     & 0 & 0 & 0 & 0 & \lambda_n
          \endbmatrix
          $$
such that $W=XSX^{-1}$. Then we can find a matrix $Y\in\mathcal
M_n(\CC)$ such that $X=e^Y$ (see e.g.~\cite{Au}) and, therefore,
$W=e^YSe^{-Y}$. Consider the holomorphic mappings
$g_{j,k}:\CC\mapsto\CC$, $j=1,\ldots,n-1$, $k=j+1,\ldots,n$ such
that $g_{j,k}(0)=0$ and $g(1)=x_{j,k}$. Then we define
$f:\CC\mapsto\Omega_n$ as follows
$$
f(\lambda):=e^{\lambda Y} \bmatrix\lambda_1 & g_{1,2}(\lambda) &
g_{1,3}(\lambda)
& \ldots & g_{1,n-1}(\lambda) & g_{1,n}(\lambda)\\
          0       &\lambda_2 & g_{2,3}(\lambda) & \ldots & g_{2,n-1}(\lambda) & g_{2,n}(\lambda)\\
          \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\
          0      & 0 & 0 & \ldots &\lambda_{n-1} & g_{n-1,n}(\lambda)\\
          0     & 0 & 0 & \ldots & 0 & \lambda_n
          \endbmatrix e^{-\lambda Y}.
          $$
\end{proof}

\begin{prop}\label{prop:14} Let $F:\Omega_n\mapsto\Omega_n$ be
holomorphic. Then there is a holomorphic mapping $\tilde F:\GG_n\mapsto\GG_n$ such that
\begin{equation}
\tilde F(\Psi_n(W))=\Psi_n(F(W)),\;W\in\mathcal M_n(\CC).
\end{equation}
\end{prop}

\begin{proof} It is sufficient to show that for any $W\in\Omega_n$
\begin{equation*}
\sigma(F(W))=\sigma(F(\diag(\lambda_1,\ldots,\lambda_n))),
\end{equation*}
where
$\sigma(W)=\{\lambda_1,\ldots,\lambda_n\}$. In view of the
previous lemma there is a holomorphic mapping $f:\CC\mapsto\Omega_n$ such that $f(0)=W$,
$f(1)=\diag(\lambda_1,\ldots,\lambda_n)$.

The mapping $\Psi_n\circ F\circ f:\CC\mapsto\GG_n$ is
holomorphic. Since $\GG_n$ is bounded we get that $\Psi_n\circ
F\circ f$ is constant. In particular,
$\pi_n(\sigma(\diag(\lambda_1,\ldots,\lambda_n)))=\pi_n(\sigma(F(W))$,
which finishes the proof.
\end{proof}

For a finite set $\emptyset\ne P\subset D$ denote $g_D(P,\cdot)$
the pluricomplex Green function with the pole set in $P$ (see
e.g.~\cite{Edi-Zwo}). If $P=\{p\}$ then we put
$g_D(p,\cdot)=g_D(\{p\},\cdot)$.

\begin{prop}\label{prop:15} Let $W_1,W_2\in \Omega_n$. Then
\begin{equation*}
g_{\Omega_n}(W_1,W_2)=g_{\Omega_n}(W_1,\diag(\lambda_1,\ldots.\lambda_n)),
\end{equation*}
where $\sigma(W_2)=\{\lambda_1,\ldots,\lambda_n\}$.
\end{prop}

\begin{proof} Let $f:\CC\mapsto\Omega_n$ be such that
$f(0)=W_2$, $f(1)=\diag(\lambda_1,\ldots,\lambda_n)$ (use
Lemma~\ref{lem:13}). Since the function
$u:=g_{\Omega_n}(W_1,f(\cdot)): \CC\mapsto[-\infty,0)$ is
subharmonic, $u$ must be constant. Therefore,
$g_{\GG_n}(W_1,W_2)=u(0)=u(1)=g_{\Omega_n}(W_1,\diag(\lambda_1,\ldots,\lambda_n))$.
\end{proof}

A domain $D\subset\CC^n$ is called hyperconvex if there exists a
negative plurisubharmonic exhaustion $u$ of $D$, i.e.~$\{z\in D:
u(z)<-\epsilon\}\Subset D$ for any $\epsilon>0$ (see
e.g.~\cite{K}). Note that any hyperconvex domain is pseudoconvex.

We have the following result.
\begin{prop}\label{prop:16} $\GG_n$ is a hyperconvex domain for any $n\ge 1$.
\end{prop}
\begin{proof} We know that $\GG_n=\pi_n(\DD^n)$. Let
$q(z)=\max\{|z_1|,\dots,|z_n|\}$, where
$z=(z_1,\dots,z_n)\in\CC^n$. Put $u(w)=\max\log q(\pi_n^{-1}(w))$,
where $w\in\CC^n$. Note that $u_{|\GG_n}$ is a negative
plurisubharmonic exhaustion of $\GG_n$ (use Proposition 2.9.26 in
\cite{K}).
\end{proof}


The following result is a partial generalization of the main
result in \cite{Ran-W}.
\begin{thm} Let $F:\Omega_n\mapsto\Omega_n$ be a proper
holomorphic mapping. Then there is a finite Blaschke product $B$
such that $\sigma(F(W))=B(\sigma(W))$, $W\in \Omega_n$.
\end{thm}

\begin{proof} Let $\tilde F:\GG_n\mapsto\GG_n$ be as in
Proposition~\ref{prop:14}. Because of Theorem~\ref{thm:1} it is
sufficient to show that $\tilde F$ is proper. It is sufficient to
show that for any sequence $\GG_n\supset(z^{\nu})$ such that
$z^{\nu}\to\partial\GG_n$ there is a subsequence $(z_{\nu_k})$
such that $\tilde F(z^{\nu_k})\to\partial\GG_n$. Fix such a
sequence $(z^{\nu})$. Let $(W^{\nu})\subset\Omega_n$ be such a
sequence that $\Psi_n(W^{\nu})=z^{\nu}$.  Let $Q\in\Omega_n$ be
any point such that $Q\not\in F(\{\det F^{\prime}=0\})$. Let $N$
denote the multiplicity of the proper mapping $F$. Denote
$\{P_1,\ldots,P_N\}=F^{-1}(Q)$. Denote
$\sigma(F(W^{\nu})):=\{\lambda_1^{\nu},\ldots,\lambda_n^{\nu}\}$.
Then because of Proposition~\ref{prop:15} and the behaviour of the
Green function under proper holomorphic mappings (see
\cite{Edi-Zwo}) we get
\begin{multline}
g_{\Omega_n}(Q,\diag(\lambda_1^{\nu},\ldots,\lambda_n^{\nu}))=
g_{\Omega_n}(Q,F(W^{\nu}))=\\
g_{\Omega_n}(\{P_1,\ldots,P_N\}, W^{\nu})\geq
\sum_{j=1}^{N}g_{\Omega_n}(P_j,W^{\nu})\geq\sum_{j=1}^Ng_{\GG_n}(\Psi_n(P_j),z^{\nu}).
\end{multline}
Now the hyperconvexity of $\GG_n$ and the convergence
$z^{\nu}\to\partial\GG_n$ imply that the last expression tends to
$0$ as $\nu\to\infty$ (see e.g.~\cite{K}).

Choosing a subsequence, if necessary, we may assume that
$\lambda_j^{\nu}\to\lambda_j$, $j=1,\ldots,n$, where
$|\lambda_j|\leq 1$. Now the assumption $|\lambda_j|<1$,
$j=1,\ldots,n$, would imply, because of the upper-semicontinuity
of the Green function, that
$g_{\Omega_n}(Q,\diag(\lambda_1,\ldots,\lambda_n))=0$ --
contradiction. Consequently, $\tilde
F(z^{\nu})=\Psi_n(W^{\nu})=\Psi_n(\diag(\lambda_1^{\nu},\ldots,\lambda_n^{\nu}))\to
\Psi_n(\diag(\lambda_1,\ldots,\lambda_n))\in\partial\GG_n$.
\end{proof}

%\begin{remark} It is known that $\tilde k_{\GG_2}=c_{\GG_2}$.
%In particular, this means that $\log\tilde k_{\GG_2}^*=g_{\GG_2}$.
%
%?????????????????? Let $r(W)$ denote the multiplicity of the
%mapping $\Psi_n(\cdot)-\Psi(W)$ at $W$. The following formula
%holds for the Green function in the domain $\Omega_n$:
%$$
%g_{\Omega_n}(P,W)=\frac{1}{r(P)}
%g_{\Omega_n}(\Psi_n(P),\Psi_n(W)).
%$$
%??????????????????????????????
%
%
%
%\end{remark}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibliographystyle{amsplain}
\begin{thebibliography}{10}

\bibitem{Abr-Fis} M.~B.~Abrahamse, S.~D.~Fisher, \textit{Mapping
intervals to intervals}, Pacific J. Math., 91 (1980), 13--27.

\bibitem{Agl-You1} J.~Agler and N.~J.~Young, \textit{A Schwarz lemma
for the symmetrized bidisc}, Bull.~London Math.~Soc., 33 (2001),
175--186.

\bibitem{Agl-You2} J.~Agler and N.~J.~Young, \textit{The hyperbolic
geometry of the symmetrized bidisc}, preprint, (2003).

\bibitem{Agl-You3} J.~Agler and N.~J.~Young, \textit{The two-by-two
spectral Nevannlina-Pick problem}, Trans.~Amer.~Math.~Soc. 256
(2004), 573--585.

\bibitem{Au} B.~Aupetit, \textit{A primer on spectral theory},
Springer Verlag, New York, 1991.

\bibitem{Bell} S.~R.~Bell, \textit{The Bergman kernel function
and proper holomorphic mappings}, Trans.~Amer.~Math.~Soc., 270
(1982), 685--691.

\bibitem{Cos1} C.~Costara, \textit{The symmetrized bidisc and
Lempert's theorem}, Bull.~London Math.~Soc., to appear.

\bibitem{Cos} C.~Costara, \textit{On the $2\times 2$ spectral
Nevanlinna-Pick problem}, preprint (2004).


\bibitem{Edi} A.~Edigarian, \textit{Proper holomorphic
self-mappings of the symmetrized bidisc}, preprint (2003).

\bibitem{Edi-Zwo} A.~Edigarian, W.~Zwonek, \textit{Invariance of
the pluricomplex Green function under proper mappings with
application}, Complex Variables, 35 (1998), 367--380.

\bibitem{Jar-Pfl3} M.~Jarnicki \& P.~Pflug,
\textit{Invariant Distances and Metrics in Complex Analysis}, de
Gruyter Expositions in Mathematics 9, Walter de Gruyter, 1993.

\bibitem{Jar-Pfl2} M.~Jarnicki \& P.~Pflug,
\textit{On automorphisms of the symmetrized bidisc},
preprint (2003).

\bibitem{Jar-Pfl4} M.~Jarnicki \& P.~Pflug,
\textit{Invariant Distances and Metrics in Complex Analysis --
revisited}, preprint.

\bibitem {K} M.~Klimek,
\textit{Pluripotential Theory}, London Math. Soc. Monographs, 6,
Clarendon Press, 1991.

\bibitem{N} R.~Narasimhan, \textit{Several Complex Variables},
Chicago Lectures in Mathematics, 1971.

\bibitem{Pfl-Zwo} P.~Pflug and W.~Zwonek,
\textit{Description of all complex geodesics in
the symmetrized bidisc}, preprint (2003).

\bibitem{Ran-W} T.~J.~Ransford and M.~C.~White, \textit{Holomorphic
self-maps of the spectral unit ball}, Bull.~London Math.~Soc., 23
(1990), 256--262.

\bibitem{Rud} W.~Rudin, \textit{Proper holomorphic mappings maps and finite
reflection groups}, Indiana Univ.~Math.~J., 31 (1982), 701--720

\bibitem{You} R.~Younis, \textit{Interpolation by a finite
Blaschke product}, Proc.~Amer.~Math.~Soc., 78 (1980), 451--452.
\end{thebibliography}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}

math0402033.extracted_bib
\begin{thebibliography}{10}

\bibitem{Abr-Fis} M.~B.~Abrahamse, S.~D.~Fisher, \textit{Mapping
intervals to intervals}, Pacific J. Math., 91 (1980), 13--27.

\bibitem{Agl-You1} J.~Agler and N.~J.~Young, \textit{A Schwarz lemma
for the symmetrized bidisc}, Bull.~London Math.~Soc., 33 (2001),
175--186.

\bibitem{Agl-You2} J.~Agler and N.~J.~Young, \textit{The hyperbolic
geometry of the symmetrized bidisc}, preprint, (2003).

\bibitem{Agl-You3} J.~Agler and N.~J.~Young, \textit{The two-by-two
spectral Nevannlina-Pick problem}, Trans.~Amer.~Math.~Soc. 256
(2004), 573--585.

\bibitem{Au} B.~Aupetit, \textit{A primer on spectral theory},
Springer Verlag, New York, 1991.

\bibitem{Bell} S.~R.~Bell, \textit{The Bergman kernel function
and proper holomorphic mappings}, Trans.~Amer.~Math.~Soc., 270
(1982), 685--691.

\bibitem{Cos1} C.~Costara, \textit{The symmetrized bidisc and
Lempert's theorem}, Bull.~London Math.~Soc., to appear.

\bibitem{Cos} C.~Costara, \textit{On the $2\times 2$ spectral
Nevanlinna-Pick problem}, preprint (2004).


\bibitem{Edi} A.~Edigarian, \textit{Proper holomorphic
self-mappings of the symmetrized bidisc}, preprint (2003).

\bibitem{Edi-Zwo} A.~Edigarian, W.~Zwonek, \textit{Invariance of
the pluricomplex Green function under proper mappings with
application}, Complex Variables, 35 (1998), 367--380.

\bibitem{Jar-Pfl3} M.~Jarnicki \& P.~Pflug,
\textit{Invariant Distances and Metrics in Complex Analysis}, de
Gruyter Expositions in Mathematics 9, Walter de Gruyter, 1993.

\bibitem{Jar-Pfl2} M.~Jarnicki \& P.~Pflug,
\textit{On automorphisms of the symmetrized bidisc},
preprint (2003).

\bibitem{Jar-Pfl4} M.~Jarnicki \& P.~Pflug,
\textit{Invariant Distances and Metrics in Complex Analysis --
revisited}, preprint.

\bibitem {K} M.~Klimek,
\textit{Pluripotential Theory}, London Math. Soc. Monographs, 6,
Clarendon Press, 1991.

\bibitem{N} R.~Narasimhan, \textit{Several Complex Variables},
Chicago Lectures in Mathematics, 1971.

\bibitem{Pfl-Zwo} P.~Pflug and W.~Zwonek,
\textit{Description of all complex geodesics in
the symmetrized bidisc}, preprint (2003).

\bibitem{Ran-W} T.~J.~Ransford and M.~C.~White, \textit{Holomorphic
self-maps of the spectral unit ball}, Bull.~London Math.~Soc., 23
(1990), 256--262.

\bibitem{Rud} W.~Rudin, \textit{Proper holomorphic mappings maps and finite
reflection groups}, Indiana Univ.~Math.~J., 31 (1982), 701--720

\bibitem{You} R.~Younis, \textit{Interpolation by a finite
Blaschke product}, Proc.~Amer.~Math.~Soc., 78 (1980), 451--452.
\end{thebibliography}