Spectral properties for some extensions of isometric operators

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 TX=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 R(E)=N(T-lambda)=N(T-lambda). Further, we show that quasi-2-isometric operator satisfies Bishop's property (beta). Finally, we prove Weyl type theorems for f(dTS), where dTS denote the generalized derivation or the elementary operator with quasi-2-isometric operator entries T and S and f is an element of H(sigma(dTS)), the set of analytic functions which are defined on an open neighborhood of sigma(d(TS)).