Spectral properties for some extensions of isometric operators

Abstract

In this paper, we show that the spectrum, Weyl spectrum, and Browder spectrum are continuous on the set of all 2-isometric operators. We also prove that if T, $S^{*}$ are 2-isometric and invertible isometric operators respectively and X is a Hilbert–Schmidt operator such that $TX=XS$, then $T^{*}X=XS^{*}$. Moreover, we show that every Riesz projection E with respect to a non-zero isolated spectral point $\lambda$ of a 2-isometric operator T is self-adjoint and satisfies ${\mathcal {R}}(E) = {\mathcal {N}}(T-\lambda ) = {\mathcal {N}}(T- \lambda )^{*}$. Further, we show that quasi-2-isometric operator satisfies Bishop’s property ($\beta$). Finally, we prove Weyl type theorems for $f(d_{TS})$, where $d_{TS}$ denote the generalized derivation or the elementary operator with quasi-2-isometric operator entries T and $S^{*}$ and $f \in H(\sigma (d_{TS}))$, the set of analytic functions which are defined on an open neighborhood of $\sigma (d_{TS})$.