We define a new generalized inverse (named the gDMP inverse) for a Hilbert space operator using its generalized Drazin inverse and its Moore–Penrose inverse. Thus, we extend the DMP inverse for a square matrix to more general case. Also, we introduce two new classes of operators, -EP and -normal operators which include, respectively, EP operators and normal operators. A new binary relation is associated with the gDMP inverse is presented and studied. The notion of core-EP inverse for matrices is extended to generalized Drazin invertible operators on Hilbert space.
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Dijana Mosić, Dragan S. Djordjević, The gDMP inverse of Hilbert space operators. J. Spectr. Theory 8 (2018), no. 2, pp. 555–573DOI 10.4171/JST/207