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

2013/07/09 by Sebastian Zaigler, Domenico P. L. Castrigiano, Zaigler, Sebastian +1
Mathematics · #41A10 #47A16 #47B15 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:41A10 #msc:47A16 #msc:47B15

paper · pdf · doi:10.48550/arxiv.1307.2437

10 pages, 0 figures

arxiv created 2013/07/09 · arxiv updated 2013/07/10

Abstract

For every, possibly unbounded, multiplication operator in L^p-space, p∈ ]0,∞[, 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 C₀(\C) and further in L^p(μ). We characterize topologically those locally compact sets X⊂ \C, for which M_z in C₀(X) is cyclic.

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