### 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^{*5}T^{5}-2T^{*3}T^{3}+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 \sigma is continuous on the class of all quasi-square-2-isometric operators.

Under the hypothesis \sigma(T)\cap (-\sigma(T))=\phi, we also prove that if E is the Riesz idempotent for an isolated point of the spectrum of quasi-square-2-isometric operator, then E is self-adjoint.

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