The aim of this paper is to introduce and study left and right versions of the class of generalized Drazin-Riesz invertible operators, as well as left and right versions of the class of generalized Drazin-meromorphic invertible operators.

Let R denote any of the following classes: upper (lower) semi-Fredholm operators, Fredholm operators, upper (lower) semi-Weyl operators, Weyl operators, upper (lower) semi-Browder operators, Browder operators. For a bounded linear operator T on a Banach space X we prove that T = T M ⊕ T N with T M ∈ R and T N quasinilpotent (nilpotent) if and only if T admits a generalized Kato decomposition (T is of Kato type) and 0 is not an interior point of the corresponding spectrum σ R (T ) = {λ ∈ C : T − λ / ∈ R}. In addition, we show that every non-isolated boundary point of the spectrum σ R (T ) belongs to the generalized Kato spectrum of T . 2010 Mathematics subject classification: 47A53, 47A10. Key words and phrases: generalized Kato decomposition, generalized Kato spectrum, upper and lower semi-Fredholm, semi-Weyl and semi-Browder operators.

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