Lp-和C0-Spaces中无界乘法算子的循环性
Cyclicity for Unbounded Multiplication Operators in Lp- and C0-Spaces
摘要：
对于每一个可能无界的，在$L^p$ -空间，$p\in ]0,\infty[$，有限可分测度空间上的乘法算子，我们证明了多重循环性，多重 *-循环性和多重性是一致的。这一结果包含并推广了1955年Bram关于有界 *-循环正规算子的被广泛引用的定理。它还包括作为核心结果的乘法运算符$M_z$乘以$L^p(μ)$中的复变量$z$的循环性，对于$\C$上的每个Borel测度$μ$。简洁的证明部分地基于以下结果：函数$e^{-|z|^2}$是在$C_0(\C)$中的$M_z$的 *-循环向量，并且进一步在$L^p(μ)$中。我们对局部紧集$X\subset \C$进行了拓扑刻画，其中$C_0(X)$中的$M_z$是循环的.
英文摘要：
For every, possibly unbounded, multiplication operator in $L^p$-space, $p\in ]0,\infty[$, on finite separable measure space we show that multicyclicity, multi-*-cyclicity, and multiplicity coincide. This result includes and generalizes Bram's much cited theorem from 1955 on bounded *-cyclic normal operators. It also includes as a core result cyclicity of the multiplication operator $M_z$ by the complex variable $z$ in $L^p(μ)$ for every Borel measure $μ$ on $\C$. The concise proof is based in part on the result that the function $e^{-|z|^2}$ is a *-cyclic vector for $M_z$ in $C_0(\C)$ and further in $L^p(μ)$. We characterize topologically those locally compact sets $X\subset \C$, for which $M_z$ in $C_0(X)$ is cyclic.
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