Some spectral properties of linear operators satisfying two important symmetric equalities

We study the importance of Drazin invertibility of operators on Banach space, in this work we present some common spectral properties for some class of bounded linear operators on a Banach space which satisfying two importante equalities between the product of the operator T and S and the power of T and S and the squart root of T and S, we We arrived at very good results which concern not only the spectra of these operators, but also different types of spectra such as the point spectrum, the approximate spectrum and other types, also we can generalize this importante result to the higher and fractional power. Drazin invertibility and its use in different domains of engineering are clear from these results obtained in our paper. We present new additive properties of the product of Drazin inverse of a linear operator on a Banach space. The Drazin invertibility of certain some operator on a Banach space is thereby established. These results extend many known results in the domain of operator theory, operator equation and the Drazin and generalized invertibilty of operator.