Cyclicity for unbounded multiplication operators in l^p- and c₀ -spaces

For every, possibly unbounded, multiplication operator in Lpspace, p 2]0,1[, 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 α-finite 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 Mz in C₀(C) and further in Lp(μ). We characterize topologically those locally compact sets X C, for which Mz in C₀(X) is cyclic.