Dynamical properties of Multiplication Operators on the Zygmund Space

Abstract

In this paper, firstly we establish bounds on the norm of multiplication operators on the Zygmund space of the unit disk via weighted composition operators.

Also then the power boundedness and mean ergodicity of multiplication operators are investigated on the Zygmund space $\mathcal{Z}$ and the little Zygmund space $\mathcal{Z}_{0}$. We completely characterize power bounded, mean ergodic and uniformly mean ergodic multiplication operators on $\mathcal{Z}$ and $\mathcal{Z}_{0}$.

Lastly we describe the spectrum of the multiplication operators in terms of the range of the symbol.