On some classes of Saphar type operators

In this paper we define and study operators of Saphar type in a Banach space, their subclasses, essentially left (right) Drazin invertible operators and left (right) Weyl–Drazin invertible operators, by means of kernels and ranges of powers of an operator. A bounded linear operator T on a Banach space X is said to be of Saphar type if T is a direct sum of a Saphar operator and a nilpotent one. We prove that T is essentially left (right) Drazin invertible if and only if T is a direct sum of a left (right) Fredholm operator and a nilpotent one, as well as that T is left (right) Weyl–Drazin invertible if and only if T is a direct sum of a left (right) Weyl operator and a nilpotent operator. We also consider the corresponding spectra.