Annals of Functional Analysis

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

Domenico P. L. Castrigiano and Sebastian Zaigler

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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.

Article information

Ann. Funct. Anal., Volume 6, Number 2 (2015), 33-48.

First available in Project Euclid: 19 December 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47A16: Cyclic vectors, hypercyclic and chaotic operators
Secondary: 41A10: Approximation by polynomials {For approximation by trigonometric polynomials, see 42A10} 47B15: Hermitian and normal operators (spectral measures, functional calculus, etc.)

Bram's theorem unbounded normal operator multiplicity cyclic vector uniform approximation


Zaigler, Sebastian; Castrigiano, Domenico P. L. Cyclicity for Unbounded Multiplication Operators in $L^p$- and $C_0\,$-Spaces. Ann. Funct. Anal. 6 (2015), no. 2, 33--48. doi:10.15352/afa/06-2-4.

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