On the $g_{z}$-Kato decomposition and generalization of Koliha Drazin invertibility

Abstract

In [24], Koliha proved that T ∈ L(X) (X is a complex Banach space) is generalized Drazin invertible operator equivalent to there exists an operator S commuting with T such that ST S = S and σ(T 2 S − T ) ⊂ {0} which is equivalent to say that 0 ∈ acc σ(T ). Later, in [14,34] the authors extended the class of generalized Drazin invertible operators and they also extended the class of pseudo-Fredholm operators introduced by Mbekhta [27] and other classes of semi-Fredholm operators. As a continuation of these works, we introduce and study the class of gz-invertible (resp., gz-Kato) operators which generalizes the class of generalized Drazin invertible operators (resp., the class of generalized Kato-meromorphic operators introduced by Živković-Zlatanović and Duggal in [35]). Among other results, we prove that T is gz-invertible if and only if T is gz-Kato with p(T ) = q(T ) < ∞ which is equivalent to there exists an operator S commuting with T such that ST S = S and acc σ(T 2 S − T ) ⊂ {0} which in turn is equivalent to say that 0 ∈ acc (acc σ(T )). As application and using the concept of the Weak SVEP introduced at the end of this paper, we give new characterizations of Browder-type theorems.closed T -invariant subspaces andan upper or a lower (resp., upper and lower) semi-B-Fredholm then T it is called semi-B-Fredholm (resp., B-Fredholm) and its index is defined by ind(T ) := α(T [mT ] ) − β(T [mT ] ). T is said to be an upper semi-B-Weyl (resp., a lower semi-B-Weyl, B-Weyl, left Drazin invertible, right Drazin invertible, Drazin invertible) if T is an upper semi-B-Fredholm with ind(T ) ≤ 0 (resp., T is a lower semi-B-Fredholm with ind(T ) ≥ 0, T is a B-Fredholm with ind(T ) = 0, T is an upper semi-B-Fredholm 0 2020