Classes of operators related to 2-isometric operators

Abstract

We introduce the class of quasi-square-2-isometric operators on a complex separable Hilbert space. This class extends the class of 2-isometric operators due to Agler and Stankus. An operator T is said to be quasi-square-2-isometric if T*5T5 ? 2T*3T3 + T*T = 0. In this paper, we give operator matrix representation of quasi-square-2-isometric operator in order to obtain spectral properties of this operator. In particular, we show that the function ? is continuous on the class of all quasi-square-2-isometric operators. Under the hypothesis ?(T)?(??(T)) = ?, we also prove that if ET({?}) is the Riesz idempotent for an isolated point of the spectrum of quasi-square-2-isometric operator, then ET({?}) is self-adjoint.