Geodesic
On the ɡz -Kato decomposition and generalization of Koliha Drazin invertibility
Filomat, Tome 37 (2023) no. 7, p. 2087

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In [24], Koliha proved that T ∈ L(X) (X is a complex Banach space) is generalized Drazin invertible operator iff there exists an operator S commuting with T such that STS = S and σ(T2S−T) ⊂ {0} iff 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 ɡz- invertible (resp., 1z-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 1z-invertible iff T is 1z-Kato with p˜(T) = q˜(T) ∞ iff there exists a commuting operator S with T such that STS = S and acc σ(T2S − T) ⊂ {0} iff 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.
DOI : 10.2298/FIL2307087A
Classification : 47A10, 47A11, 47A15, 47A25, 47A53, 47A55
Keywords: ɡz-Kato decomposition, ɡz-invertible operator, Weak SVEP 
Zakariae Aznay; Abdelmalek Ouahab; Hassan Zariouh. On the ɡz -Kato decomposition and generalization of Koliha Drazin invertibility. Filomat, Tome 37 (2023) no. 7, p. 2087 . doi: 10.2298/FIL2307087A
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     title = {On the ɡz {-Kato} decomposition and generalization of {Koliha} {Drazin} invertibility},
     journal = {Filomat},
     pages = {2087 },
     year = {2023},
     volume = {37},
     number = {7},
     doi = {10.2298/FIL2307087A},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/FIL2307087A/}
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