Spectral continuity and the dynamics of quasi-2-isometric operators

Linear dynamics has been a rapidly evolving area since the early 1990s. It lies at the intersection of operator theory and topological dynamics and its central property is hypercyclycity. The class of quasi-2-isometric operators on Hilbert space extends the classes of 2-isometric operators due to Agler and Stankus and quasi-isometries by S. M. Patel. An operator T on a complex Hilbert space is called quasi-2-isometry if \(\begin{aligned} T^{*3}T^{3 }- 2T^{*2 }T^{2}+ T^{*}T = 0. \end{aligned}\) $$\begin{array}{c}\hfill T^{\ast 3}{T}^3-2T^{\ast 2}{T}^2+T^{\ast }T=0.\end{array}$$ In the present article, we prove that a weakly supercyclic quasi-2-isometric operator is a unitary operator and a quasi-2-isometric operator is not weakly hypercyclic. We also, show that the spectrum is continuous on the class of all quasi-2-isometric operators and Weyl’s theorem holds for quasi-2-isometric operators.