On a class of Drazin invertible operators for which $$\left( S^{*}\right) ^{2}\left( S^{D}\right) ^{2}=\left( S^{*}S^{D}\right) ^{2}$$

In this paper we introduce and analyze a new class of operators for which
for a bounded linear operator S acting on a complex Hilbert space
using the Drazin inverse
of S. After establishing the basic properties of such operators. We show some results related to this class on a Hilbert space. In addition, we characterize the direct sum and the tensor product of these operators. At the end of this paper we generalize the Kaplansky theorem’s to this class.