Property ($gR$) under nilpotent commuting perturbation

Abstract

The property ($gR$), introduced in [Aiena, P.,Guillen, J. and Pe\~{n}a, P., {ıt Property ($gR$) andperturbations}, to appear in Acta Sci. Math. (Szeged), 2012], isan extension to the context of B-Fredholm theory, of property($R$), introduced in [Aiena, P., Guillen, J. and Pe\~{n}a, P.,{ıt Property ($R$) for bounded linear operators}, Mediterr. J.Math. {\bf 8} (4), 491-508, 2011]. In this paper we continue thestudy of property ($gR$) and we consider its preservation underperturbations by finite rank and nilpotent operators. We alsoprove that if $T$ is left polaroid (resp\. right polaroid) and $N$is a nilpotent operator which commutes with $T$ then $T+N$ is alsoleft polaroid (resp\. right polaroid).