Abstract

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

Keywords: Property ($gR$); semi B-Fredholm operator; perturbation.

MSC: 47A10, 47A11, 47A53, 47A55

Pages:  140$-$147     

Volume  66 ,  Issue  2 ,  2014