Cyclicity for Unbounded Multiplication Operators in Lp- and C0-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 Borel measure $\mu$ 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 $L^p(\mu)$. We characterize topologically those locally compact sets $X\subset \C$, for which $M_z$ in $C_0(X)$ is cyclic.