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\begin{document}

\bc
{\bf A GENERALISATION OF A THEOREM
OF KOLDUNOV WITH AN ELEMENTARY PROOF}
\ec

\mb \vs{0.1cm}

\bc
{\bf Zafer ERCAN \footnote{This work has been partially supported
by the Scientific and Technical Research Council of Turkey\\
1991 Mathematics Subject Classification. 46A40, 47H30\\
Key words and phrases. Riesz spaces (vector lattices),
Hammerstein property and disjointness preserving
operators.}}
\ec


\bc
{\bf ABSTRACT}
\ec

In this paper it is proved that disjointness preserving quasi-linear
operators between Riesz spaces has the Hammerstein propert, 
which is a generalisation of a theorem of Koldunov in [2].

Throughout this paper an operator $T:E \ra F$ between Riesz spaces
$E, F$ (not necessarily the Archimedean property) means a function
from $E$ into $F$. Let $T: E \ra F$ be an operator from a Riesz
space $E$ into another Riesz space $F$. We call $T$ positive if
$T(x) \leq T(y)$ for all $x,y \in E$ with $x \leq y$. We say that $T$
is order bounded if $T([x,y])$ is an order bounded subset of $F$
for all $x,y \in E$. $T$ is said to be disjointness preserving
(or equivalently, $T$ preserves disjointness) if $\mid T(x)\mid
\wedge \mid T(y)\mid =0$ in $F$ for all $x,y \in E$ with
$\mid x\mid \wedge \mid y \mid =0$ in $E$. For unexplained definitions
and notations we refer to ([2], [3] and [4]).

Koldunov ([2]) has modified  {\em the Hammerstein} and
{\em weak Hammerstein} property for operators between Riesz
spaces as follows:

{\bf Definition 1. ([2])} Let $T: E \ra F$ be an operator
between Riesz space $E$ and $F$. Consider the following
statements:

\begin{enumerate}
\item[(i)] $T (u+v+x)-T(u+x)=T(v+x)-T(x)$ for all disjoint
$u,v \in E$ and for all $x\in E$.
\item[(ii)] $\mid T(u+v+x)-T(u+x)\mid \leq \mid T(v+x)\mid
+ \mid T(x)\mid$ for all disjoint $u, v \in E$ and for all
$x \in E$.
\end{enumerate}
We say that $T$ satisfies the {\em weak Hammerstein property 
(the Hammerstein property)} whenever it satisfies statement
(ii) (statement i)), respectively. $\Box$

It is obvious that operators with the Hammerstein property
($\Ra$ weak Hammerstein) are disjointly additive and each
linear operator satisfies the Hammersteion property. It makes
sense as operators with the Hammerstein property may be
viewed as a natural generalisation of linear operators on the
Riesz spaces. The following characterisation of operators with
the Hammersteion property is given in [1]. We repeat it below
for it to be self-contained of this paper.

{\bf Theorem 1. ([1])} Let $T: E \ra F$ be an operator between
Riesz spaces $E$ and $F$. The following are equivalent:
\begin{enumerate}
\item[(i)]  $T$ satisfies the Hammerstein property
\item[(ii)] $T(x)+T(y)= T(x \vee y) + T(x\wedge y)$ for all $x,y
\in E$.
\end{enumerate}

{\bf Proof:} First assume that (i) holds. It follows from the
following equality.
\[x\vee y = (x-y)^+ + (x-y)^- + (y-(x-y)^-)\]
and from the disjointness of $(x-y)^+$ and $(x-y)^-$ that
$T(x)+ T(y) = T(x\vee y)+ T (x \wedge y)$ for all $x, y \in E$,
holds. Now suppose that (ii) holds. Let $u,v, x \in E$ 
beg iven in which $u,v$ are disjoint. First suppose that
$u, v \geq 0$. Then the following equalities.
\[ u+ v+x= (u+x)\vee (v+x)
\quad \mbox{and}\quad
(u+x) \wedge (v+x)=x\]
imply that
\[ T(u+v+x)-T(u+x) = T(v+x)-T(x)\]
For general cases, letting $y=x^- - v^-$ and with an easy
processing it is easy to show that
\[T(u+v+x)- T(u+x) = T(v^+ +x)-T(v^- +y) = T(v+x) - T(x)\]
This cmopletes the proof $\Box$

Koldunov ([2]) has defined quasi-linear operators between
Riesz spaces as follows:

{\bf Definition 2. ([2])} Let $T: E\ra F$ be an operator
between Riesz spaces $E$ and $F$. Consider the following
statements.
\begin{enumerate}
\item[(i)] for each $w \in E$ there exists a real number
$\lam (w)>0$ such that
\[ \mid T(u) - T(v) \mid  \leq \lam (w) \mid T(u-v)\mid \]
whenever $\mid u \mid, \mid v \mid \leq \mid w\mid$
\item[(ii)] for each $u \in E$ and $r\in \R$ there exists
real numbers $k(r,u) >0$ and $l(r,u)>0$ such that for
each $v \in [- \mid u \mid, \mid v \mid ]$,
\[l (r,u) \mid T(v) \mid \leq \mid T(rv) \mid \leq
k(r,u)\mid T(v)\mid\]
and also $\lim k(r,u)=0$ as $r \ra 0$.
\end{enumerate}
We say   that $T$ is a {\em weak quasi-linear (quasi-linear)}
whenever $T$ satisfies the statement (i) (the statements (i)
and (ii)), respectively $\Box$

Koldunov ([2], Theorem 2.6) has proved that an 
$(r_u, -r_u)$-continuous operator $T:E \ra F$ between
Archimedean Riesz spaces $E$ and $F$ has the Hammerstein
property whenever it satisfies the $DQLH$ (=disjointness
preseving, quasi-linear satisfying the weak Hammerstein)
property. Although, Koldunov's proof is based on representation
theory and quite difficult we can generalise his theorem
with an easy and direct proof as follows:

{\bf Theorem 2.} Let $T: E\ra F$ be a disjointness preserving,
weak quasi-linear operator between Riesz spaces $E$ and $F$
(not necessarily Archimedean). Then $T$ satisfies the Hammerstein
property.

{\bf Proof.} Let $x,y \in E$ be given. Then we have that,
\begin{eqnarray*}
\mid T(x)-T(x\vee y)+ T(y) - T(x\wedge y)\mid
&\leq & \mid T(x) -T(x\vee y)\mid\\
& + &\mid T(y) - T(x\wedge y)\mid\\
&\leq&  2 \lam \mid T(y-x)^+\mid 
\end{eqnarray*}
and similarly 
\begin{eqnarray*}
\mid T(x)-T(x\vee y)+ T(y) - T(x\wedge y)\mid
&\leq &\mid T(x) -T(x\vee y)\mid\\
& + &\mid T(y) - T(x\vee y)\mid\\
&\leq& 2 \lam \mid T(x-y)^+\mid 
\end{eqnarray*}
where $\lam =\lam (w)>0$, $w=\mid x\mid \vee \mid y\mid$
(the same notation as in the definition 2). Since $T$ is
disjointness preserving $\mid T(x-y)^+ \mid \wedge \mid
T(y-x)^+\mid =0$, this implies that $T(x) + T(y)=
T(x\vee y)+ T(x\wedge y)$ for all $x,y \in E$. Now by theorem 1,
$T$ has the Hammerstein property $\Box$

When we restrict ourself to the order bounded operators we can 
generalise another theorem of Koldunov ([2], theorem 3.4) as
follows:

{\bf Theorem 3.} Let $T:E\ra F$ be an order bounded quasi-linear
operator between Riesz spaces $E$ and $F$. Then $T$ is
$(r_u - r_u)$-continuous. In particular, if $F$ is Archimedean and
$T$ is also disjointness preserving, then $T$ is:

\[ additive \quad \Lra\quad homogeneous \quad \Lra \quad Linear\]

{\bf Proof.} Suppose that $x_n \ra x(r \cdot u)$ in $E$.
Then there exists $0\leq u \in E$ such that $x_n \ra
x(u)$ i.e. for each $\ep >0$ there exists a natural number
$n_0$ such that $\mid x_n -x\mid \leq \ep u$ for all
$n \geq n_0$. Let $\ep > 0$. Letting $\lam = \lam (u+ \mid x \mid )$
(the same notation as in definition 2) we have that
\begin{eqnarray*}
\mid T(x_n) - T(x) & \leq & \lam T(x_n -x)\mid\\
& \leq & \lam T(\ep (\ep^{-1} (x_n -x)\mid\\
& \leq & \lam k (\ep, u + \mid x\mid ) \mid (\ep^{-1}
(x_n -x))\mid\\
& \leq & \lam k (\ep, u+ \mid x\mid )y
\end{eqnarray*}
for all $n \geq n_0$, where $v$ is an upper bound of
$T[-u,u]$ in $F$. Since $k(\ep, u+\mid x\mid)\ra 0$ as
$\ep \ra 0$ $T(x_n)\ra T(x) (r \cdot u)$. Hence $T$
is $(r_u -r_u)$-continuous. For the rest of the proof,
combine the theorem 2 and ([3], theorem 2.6) $\Box$

Let $T:E\ra F$ be an operator between Riesz space 
$E$ and $F$ and suppose that $T(x)- T(y) \in A_{T(x-y)}$
(=the ideal generated by $T(x-y))$ for all $x,y \in E$.
If $T$ is also a disjointness preserving operator then it
is clear that $T(x) - T(x\wedge y)$ and $T(y) - T(x\wedge y)$
are disjoint for all $x,y \in E$. Then it is easy to see that
$T(x\vee y)\leq T(x) \vee T(y)$ and $T(x) \wedge T(y)
\leq T(x\wedge y)$ whenever $T$ is disjointness preserving
weak quasi-linear. It leads to the following Theorem.

{\bf Theorem 4.} Let $T:E \ra F$ be a disjointness preserving
weak quasi-linear operator between Riesz spaces $E$ and $F$.
Then the following are equivalent:
\begin{enumerate}
\item[(i)] $T$ is positive
\item[(ii)] $T$ is lattice homomorphism, i.e., $T(x\vee y)=
T(x) /vee T(y)$ and $T(x\wedge y)= T(x) \wedge T(y)$ for
all $x,y \in E$.
\end{enumerate}

\bc
{\bf References}
\ec
\begin{enumerate}
\item[{[1]}] Z.Ercan and A.W. Wickstead, Towards a theory
of non-linear orthomorphisms (submitted).
\item[{[2]}] A. V. Koldunov, Hammerstein operators preserving
disjointness, Proc. Amer. Math. Soc. 4(1995), 1083-1095.
\item[{[3]}] W. A. J. Luxemburg and A. C. Zaanen, Riesz
Spaces 1, North-Holland, Amsterdam, 1971.
\item[{[4]}] P.Meyer-Nieberg, Banach Lattices, Springer
Universitest, Berlin, 1992.
\end{enumerate}

\begin{flushleft}
Auther's address: Department of Mathematics,\\
Middle East Technical University\\
06531 Ankara, Turkey\\
e-mail:zercan@rorqual.cc.metu.edu.tr.
\end{flushleft}
\end{document}