Cyclicity for Unbounded Multiplication Operators in Lp- and C0-Spaces

TL;DR

This paper proves that for unbounded multiplication operators in Lp-spaces, multicyclicity, multi-*-cyclicity, and multiplicity are equivalent, extending classical results and characterizing cyclicity in various function spaces.

Contribution

It generalizes Bram's 1955 theorem to unbounded operators and characterizes cyclicity of multiplication operators in Lp and C0 spaces.

Findings

Multicyclicity, multi-*-cyclicity, and multiplicity coincide for unbounded multiplication operators.

Cyclicity of M_z in Lp spaces is established for all Borel measures on C.

Characterization of topological sets where M_z is cyclic in C0(X).

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.