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URN etd-0127105-231944 Author Ying-Fen Lin Author's Email Address linyf@math.nsysu.edu.tw Statistics This thesis had been viewed 5169 times. Download 1928 times. Department Applied Mathematics Year 2004 Semester 1 Degree Ph.D. Type of Document Language English Title Disjointness preserving operators on function spaces Date of Defense 2005-01-19 Page Count 57 Keyword compact operator weakly compact operator disjointness preserving operator completely continuous operator spectrum Abstract 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.Advisory Committee Pei Yuan Wu - chair

Mark Ho - co-chair

Jen-Chih Yao - co-chair

Mau-Hsiang Shih - co-chair

Chin-Cheng Lin - co-chair

Sen-Yen Shaw - co-chair

Ngai-Ching Wong - advisor

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etd-0127105-231944.pdf Date of Submission 2005-01-27