Eigenvalues and dynamical properties of weighted backward shifts on the space of real analytic functions
Volume 242 / 2018
Studia Mathematica 242 (2018), 57-78
MSC: Primary 47B37, 46E10, 47A16; Secondary 47A10.
DOI: 10.4064/sm8739-6-2017
Published online: 11 January 2018
Abstract
Usually backward shift is neither chaotic nor hypercyclic. We will show that on the space $\mathscr {A}(\varOmega )$ of real analytic functions on a connected set ${\varOmega }\subseteq \mathbb {R}$ with $0\in {\varOmega }$, the backward shift operator is chaotic and sequentially hypercyclic. We give criteria for chaos and for many other dynamical properties for weighted backward shifts on $\mathscr {A}(\varOmega )$. For special classes of them we give full characterizations. We describe the point spectrum and eigenspaces of weighted backward shifts on $\mathscr {A}(\varOmega )$ as above.
