Generalized Saphar decomposition of certain subclasses of generalized Drazin invertible linear relations

For a Banach space, the notions of essentially left and right generalized Drazin invertible linear relations are introduced and studied. Then, characterizations of these classes by means of their generalized Saphar decompositions, accumulation and interior points of various spectra are given. Furthermore, sufficient conditions under which an essentially left (resp. right) generalized Drazin invertible linear relation be left (resp. right) Weyl generalized Drazin invertible are provided. In particular, we show that an everywhere defined closed linear relation with a nonempty resolvent set which has the SVEP at \(0\) (resp. its adjoint has the SVEP at \(0\)) is essentially left (resp. right) generalized Drazin invertible if and only if it is left (resp. right) Weyl generalized Drazin invertible. The corresponding spectra of such classes are also investigated and concrete examples are illustrated.