Shen J. L.
Ukr. Mat. Zh. - 2017. - 69, № 7. - pp. 974-983
We introduce a new property $(gaR)$ extending the property $(R)$ considered by Aiena. We study the property $(gaR)$ in connection with Weyl type theorems and establish sufficient and necessary conditions under which the property $(gaR)$ holds. In addition, we also study the stability of the property $(gaR)$ under perturbations by finite-dimensional operators, by nilpotent operators, by quasinilpotent operators, and by algebraic operators commuting with $T$. The classes of operators are considered as illustrating examples.