On an elementary operator with 2-isometric operator

Abstract

A Hilbert space operator T is said to be a 2-isometric operator if T^{*2}T^{2}-2T^{*}T+I=0. Let d_{AB}\in B(B(H)) denote either the generalized derivation \delta_{AB}= L_{A}-R_{B} or the elementary operator \Delta_{AB} =L_{A}R_{B}- I, we show that if A and B^{*} are 2-isometric operators, then, for all complex \lambda, (d_{AB}-\lambda)^{-1}(0) \subseteq (d^{*}_{AB}-\overline{\lambda})^{-1}(0), the ascent of (d_{AB}-\lambda)\leq1, and d_{AB} is polaroid. Let H(\sigma(d_{AB})) denote the space of functions which are analytic on \sigma(d_{AB}), and let H_{c}(\sigma(d_{AB})) denote the space of f\in H(\sigma(d_{AB})) which are non-constant on every connected component of \sigma(d_{AB}), it is proved that if A and B^{*} are 2-isometric operators, then f(d_{AB}) satisfies the generalized Weyl's theorem and f(d^{*}_{AB}) satisfies the generalized a-Weyl's theorem.