令$X$和$Y$為局部緊緻豪士道夫(locally compact
Hausdorff)空間，且$T$是一個從$C_0(X)$到$C_0(Y)$的線性有界保斥性算子。我們給了數個使$T$為緊緻算子(compact)的等價條件，分別是：$T$是弱緊緻的(weakly compact)； $T$是完全連續的(completely continuous)； $T$可以表示成 $T= sum_n delta_{x_n} otimes h_n$的形式，這裡${x_n}_n$是在$X$上的數列，${h_n}_n$是在$C_0(Y)$上互斥且範數收斂到零的數列。對每一個滿足上述條件的緊緻保斥性算子$T$，我們可以對應到一個具有可數多個點的圖，再透過這個圖的結構，就可以完整的描述出$T$的譜(spectrum)。具體來說，一個非零的複數$lambda$是$T$的特徵值(eigenvalue) 若且唯若存在某個正整數$k$，使得$lambda$滿足$lambda^k= h_1(x_k) h_2(x_1) cdots h_k(x_{k-1})$。

令$X$和$Y$為局部緊緻豪士道夫空間，$E$和$F$為巴拿赫(Banach
space)空間，我們也給出從$C_0(X,E)$映到$C_0(Y,F)$的緊緻保斥性算子$T$的分解：$T= sum_n delta_{x_n} otimes
h_n$，其中${x_n}_n$是一個在$X$上的數列，${h_n}_n$是一個從$Y$映到$B(E,F)$互斥且範數收斂到零的數列並在均勻算子拓樸(uniform operator topology)上連續且在無窮遠點收斂到零，再者，對任意$y in Y$，$h_n(y)$是緊緻算子。對完全連續保斥性算子，我們有相似的結果。具體來說，完全連續保斥性算子可以分解為可數個完全連續分子$delta_{x_n} otimes h_n$的和，這裡$h_n: Y o B(E,F)$在強算子拓樸(strong operator topology)上連續並在無窮遠點收斂到零，而且$h_n$是一致完全連續的(uniformly completely continuous)。對弱緊緻保斥性算子$T$，只有在巴拿赫空間$E$是可分(separable)的情況下，$T$才可分解為可數多個弱緊緻分子的合。我們給了一個當$E$是不可分的情形下的反例。

Let $T$ be a bounded disjointness preserving linear operator from $C_0(X)$ into $C_0(Y)$, where $X$ and $Y$ are locally compact Hausdorff spaces. We give several equivalent conditions for $T$ to be compact; they are: $T$ is weakly compact; $T$ is completely continuous; $T= sum_n delta_{x_n} otimes h_n$ for a sequence of distinct points ${x_n}_n$ in $X$ and a norm null mutually
disjoint sequence ${h_n}_n$ in $C_0(Y)$. The structure of a
graph with countably many vertices associated to such a compact operator $T$ gives rise to a new complete description of the spectrum of $T$. In particular, we show that a nonzero complex number $la$ is an eigenvalue of $T$ if and only if $lambda^k= h_1(x_k) h_2(x_1) cdots h_k(x_{k-1})$ for some positive integer $k$.

We also give a decomposition of compact disjointness preserving operators $T$ from $C_0(X,E)$ into $C_0(Y,F)$, where $X$ and $Y$ are locally compact Hausdorff spaces, $E$ and $F$ are Banach spaces. That is, $T= sum_n de_{x_n} otimes h_n$ for a sequence of distinct points ${x_n}_n$ in $X$ and a norm null mutually disjoint sequence ${h_n}_n$, where $h_n: Y o B(E,F)$ is continuous and vanishes at infinity in the uniform operator topology and $h_n(y)$ is compact for each $y$ in $Y$. For completely continuous disjointness preserving linear operators, we get a similar decomposition. More precisely, completely continuous
disjointness preserving operators $T$ have a countable sum
decomposition of completely continuous atoms $de_{x_n} otimes h_n$, where $h_n: Y o B(E,F)$ is continuous, vanishes at infinity in the strong operator topology and $h_n$ is uniformly completely continuous. In case of weakly compact disjointness preserving linear operators, $T$ have a countable sum decomposition of weakly compact atoms whenever the Banach space $E$ is separable. A counterexample is given whenever $E$ in nonseparable.

Contents
Chapter 1: Introduction 1
Chapter 2: Compact disjointness preserving operators of continuous scalar functions 5
2.1 A structure theorem for compact disjointness preserving operators .... 5
2.2 More characterizations of compact disjointness preserving operators ....10
2.3 A spectral theory of compact disjointness preserving operators . . . . 14
Chapter 3: Disjointness preserving operators of continuous vectorvalued functions.... 25
3.1 Characterization of the operator $delta_{x_n} otimes h_n$ . . . . . . . . . . . . . . . . . 26
3.1.1 The compactness of $delta_{x_n} otimes h_n$ . . . . . . . . . . . . . . . . 26
3.1.2 The weak compactness of $delta_{x_n} otimes h_n$. . . . . . . . . . . . . . . . 27
3.1.3 The complete continuity of $delta_{x_n} otimes h_n$. . . . . . . . . . . . . . 28
3.2 Compact and Completely continuous disjointness preserving operators .... 30
3.3 Weakly compact disjointness preserving operators . . . . . . . . . . . . 35
3.4 A spectral theory of compact disjointness preserving operators . . . . 40
Chapter 4: Power compact disjointness preserving operators of continuous functions .... 44
4.1 An example of unbounded power compact disjointness preserving operators . . . . . 44
4.2 A structure theorem for power compact disjointness preserving operators .....46
4.3 Spectral theories of power compact disjointness preserving operators ..... 46
References . . . . . . . . . 51

ibitem{Abr83} Yu. A. Abramovich,
Multiplicative representation of disjointness preserving
operators, emph{Indag. Math.} { f 45, no. 3} (1983), 265--279.

ibitem{Abr79} Yu. A. Abramovich, A. I. Veksler and A. V. Koldunov, On operators preserving disjointness, emph{Soviet Math. Dokl.} extbf{248} (1979), 1033--1036.

ibitem{Ali} C. D. Aliprantis and O. Burkinshaw, emph{Positive operators}, Academic Press, Orlando, 1985.

ibitem{Are83} W. Arendt, Spectral properties of Lamperti operators, emph{Indiana Univ. Math. J.} { f 32, no. 2} (1983), 199--215.

ibitem{Bartle} R. G. Bartle, N. Dunford and J. Schwartz, Weak compactness and vector measures,
emph{Can. J. Math.} extbf{7} (1955), 289--305.

ibitem{Beh79}
E. Behrends, emph{$M$-structure and the Banach Stone Theorem}, Springer-Verlag, Berlin, 1979.

ibitem{Brown03}
L. G. Brown and N.-C. Wong,
Unbounded disjointness preserving linear functionals,
emph{Monatsh. Math.} extbf{141} (2004), 21--32.

ibitem{Chan90} Jor-Ting Chan, Operators with the disjoint support property, emph{J. Operator Theory} extbf{24} (1990), 383--391.

ibitem{Con90} J. B. Conway,
emph{A course in functional analysis, 2nd edition},
Spring-Verlag, Berlin, 1990.