Spectral properties and the dynamics of quasi-2-expansive operators

Abstract

An operator T on a complex Hilbert space is called a quasi-2-expansive operator if T^{\*3}T^{3}-2T^{\*2}T^{2}+T^{\*}T\leq0, which is a further generalization of 2-isometric operator. In this paper, some basic structural properties of quasi-2-expansive operators are established with the help of operator matrix representation. Using these results, we show that the spectrum is continuous on the class of all quasi-2-expansive operators. Moreover, we verify that Weyl's theorem holds for quasi-2-expansive operators. Finally, we prove that a weakly supercyclic quasi-2-expansive operator is a unitary operator.