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In this article, we consider Drazin invertible operators for study of the relationship between their <i>B</i>-Fredholm spectra and the transfer between some of the spectral properties defined through <i>B</i>-Fredholm spectra of this class of operators. Among other results, we investigate the transfer of generalized <i>a</i>-Weyl’s theorem from <i>T</i> to their Drazin inverse <i>S</i>, if it exists.