1$
such that
\begin{equation}
\label{hil30}
\lambda (u)u_0\leq S(u)\leq \beta \lambda (u)u_0,\quad
u\in \mathop{\rm int}K,
\end{equation}
where $\lambda (u)$ is a positive constant depending upon $u$.
We have the following theorem.
\begin{theorem} \label{perron}
Let the norm of $E$ be monotone with respect to the cone $K$ and
let $T$ be a linear positive operator satisfying (\ref{pos}) such
that for some integer $n$, the operator $T^n$
is uniformly positive. Then there exists a unique pair $(\mu ,u)\in
(0,\infty )\times \mathbb{M}$
such that
\begin{equation}
\label{hil31}
T(u)=\mu u.
\end{equation}
\end{theorem}
\begin{proof}
We define the mapping
$g:\mathbb{M}\to \mathbb{M}$ by
\[
g(u):= \frac{T(u)}{\|T(u)\|},
\]
and let
\[
f:=\underbrace{g\circ \dots \circ g}_{n},
\]
i.e., $g$ composed with itself $n$ times.
Then
\[
f(u)= \frac{S(u)}{\|S(u)\|},
\]
where $S=T^n$.
It follows from the properties of Hilbert's projective metric,
that
\[
{\rm d}(f(u),f(v))={\rm d}(S(u),S(v)),\quad u,v\in \mathbb{M},
\]
and that $\mathbb{M}$ is complete, since the norm of $E$ is monotone
with respect to the cone $K$.
Thus, $f$ will have a unique fixed point, once we show that
$f$ is a contraction mapping, which will follow from Theorem \ref{hil} once we establish that
$\Delta (S)<\infty $.
To compute $\Delta (S)$, we recall the definition of projective
diameter\index{projective diameter} (see (\ref{hil17})) and find
that for any $u,v\in \mathbb{M}$,
\[
{\rm d}(S(u),S(v))\leq {\rm d}(S(u),u_0)+{\rm d}(S(v),u_0)
\]
and, therefore, by the uniform positivity of $T^n$,
\[
{\rm d}(S(u),u_0),\;{\rm d}(S(v),u_0)\leq \log\beta ,
\]
implying that
\[
{\rm d}(S(u),S(v))\leq 2\log \beta .
\]
Thus $S$, and hence, $f$, are contraction mappings with respect to
the projective metric and therefore, there exists a unique $u\in
\mathbb{M}$ such that
\[
f(u)=u,
\]
i.e. $S(u)=u$,
or
\[
T^n(u)=\|T^n(u)\| u,
\]
and the direction $u$ is unique. Furthermore, since $f$ has a
unique fixed point in $\mathbb{M} $, $g$ will have a unique fixed
point also, as follows from Theorem \ref{pcontraction} of Chapter
\ref{chapIII}. This also implies the uniqueness of the eigenvalue
with corresponding unique eigenvector $u\in \mathop{\rm int}K,\;\|u\|=1$.
\end{proof}
In the following we provide two examples to illustrate the above
theorem.
The first example illustrates part of the Perron-Frobenius theorem and
the second is an extension of this result to operators on spaces of
continuous functions. We remark here that the second result concerns
an integral equation which is not given by a compact linear operator
(see also \cite{birkhoff:ejt57}).
\begin{example} \label{hilbexm5} \rm
Let
\[
E=\mathbb{R}^N,\quad
K=\{(u_1,u_2, \dots , u_N): u_i\geq 0,\;i=1,2, \dots , N\}.
\]
Let
$T:K\to K$
be a linear transformation whose $N\times N$ matrix representation
is irreducible. Then there exists a unique pair $(\lambda , u)\in
(0,\infty )\times \mathop{\rm int}K$, $\|u\|=1 $, such that
\[
Tu =\lambda u.
\]
\end{example}
\begin{proof}
An $N\times N$ matrix is irreducible (see \cite{leon:laa98}), provided
there does not
exist a
permutation matrix $P$ such that
\[
PTP^T=\begin{pmatrix}
B&O\\
C&D
\end{pmatrix},
\]
where $B$ and $D$ are square submatrices. This is equivalent to
saying, that for some positive integer $n$, the matrix $T^n=\left
(t_{i,j}\right )$ has only positive entries $t_{i,j},i,j=1,\dots
, N$. Since,
\[
\mathop{\rm int}K=\{(u_1,u_2,
\dots , u_N): u_i> 0,\;i=1,2, \dots , N\},
\]
if we let
\[
m=\min _{i,j}t_{i,j},\quad
M=\max _{i,j}t_{i,j},\quad u_0=(1,1,\dots ,1),
\]
then for any $u\in K^+$,
\[
m\|u\|_1u_0\leq T^nu\leq M\|u\|_1u_0,
\]
where
\[
\|u\|_1=\sum _{i=1}^N|u_i|,
\]
is the $l_1$ norm of the vector $u$.
This shows that $T^n$ is a uniformly positive operator. Hence Theorem
\ref{perron} may be applied.
\end{proof}
For many applications of positive matrices (particularly to economics)
we refer to \cite{leon:laa98}, \cite{strang:laa88}.
The following example is discussed in \cite{birkhoff:ejt57}.
Let again $E:=C[0,1]$,
with the usual maximum norm and $K$ the cone of nonnegative
functions.
Let
\[
p:[0,1]^2\to (0,\infty)
\] be a continuous function.
Let
\[
0 |t|/\eta$.
Geometrically, in the real case, the set $\Omega_t$
is the projection onto $\mathbb{C}^n$ of
the base of that part of $\Omega $ which
lies above $t$ ($t>0$) or below $t$ ($t<0$).
For $z\in \Omega_t$, we define $d(t,z)$ by
\begin{equation}\label{distform}
d(t,z):= d(z)-\frac{|t|}{\eta}.
\end{equation}
The function $d(t,z)$ is positive and represents the distance
from $z\in \Omega_t$ to $\Gamma_t$. The
following property of $d(t,z)$ will be needed
later in the proof of the theorem.
\begin{lemma}\label{distprop}
If $z'\in \mathbb{C}^n$ satisfies $|z-z'|=r