Cyclicity for Unbounded Multiplication Operators in $L^p$- and $C_0\,$-Spaces

Abstract

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 $L^p(\mu)$ for every $\sigma$-finite Borel measure $\mu$ on $\mathbb{C}$. The concise proof is based in part on the result that the function $e^{-\left|z\right|^2}$ is a $*$-cyclic vector for $M_z$ in $C_0(\mathbb{C})$ and further in $L^p(\mu)$. We characterize topologically those locally compact sets $X\subset \mathbb{C}$, for which $M_z$ in $C_0(X)$ is cyclic.