Independent Backward Preimages and Algebraic Radial Decompositions of Operators
Abstract
Let T∈ℬ(X), where X is a complex Banach space. We introduce the spaces ℐs(T), defined by independently chosen preimages under the powers of T with exponential growth bounds, and study algebraic radial decompositions involving ℋr(T), 𝒦s(T), and ℐs(T). We prove that every algebraic decomposition X=ℋr(T)⊕ℐs(T) is topological and that ℐs(T)=𝒦s(T). This removes the closedness assumption in the radial decomposition theorem of Koliha and Poon. We also prove that, for 0≤ a≤ b, the closedness of ℐb(T) and ℋa(T)+ℐb(T) implies ℐb(T)=𝒦b(T). We establish finite-defect and circular spectral criteria and characterize the Cantor--Bendixson derivatives of the spectrum, Browder spectrum, and Drazin spectrum in terms of radial decompositions and quotient restrictions. As applications, we obtain new characterizations of generalized Drazin, generalized Drazin--Riesz, generalized Drazin--meromorphic, and gz-invertible operators in terms of radial subspaces.
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BibTeX
@misc{aznay2026independent,
title = {{Independent Backward Preimages and Algebraic Radial Decompositions of Operators}},
author = {Zakariae Aznay and Fadil Chabbabi and Hassan Zariouh},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06831},
eprint = {2611.06831},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06831},
}
RIS
TY - UNPB
TI - Independent Backward Preimages and Algebraic Radial Decompositions of Operators
AU - Zakariae Aznay
AU - Fadil Chabbabi
AU - Hassan Zariouh
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06831
UR - https://arxiv.org/abs/2611.06831
AB - Let T∈ℬ(X), where X is a complex Banach space. We introduce the spaces ℐ_s(T), defined by independently chosen preimages under the powers of T with exponential growth bounds, and study algebraic radial decompositions involving ℋ_r(T), 𝒦_s(T), and ℐ_s(T). We prove that every algebraic decomposition X=ℋ_r(T)⊕ℐ_s(T) is topological and that ℐ_s(T)=𝒦_s(T). This removes the closedness assumption in the radial decomposition theorem of Koliha and Poon. We also prove that, for 0≤ a≤ b, the closedness of ℐ_b(T) and ℋ_a(T)+ℐ_b(T) implies ℐ_b(T)=𝒦_b(T). We establish finite-defect and circular spectral criteria and characterize the Cantor--Bendixson derivatives of the spectrum, Browder spectrum, and Drazin spectrum in terms of radial decompositions and quotient restrictions. As applications, we obtain new characterizations of generalized Drazin, generalized Drazin--Riesz, generalized Drazin--meromorphic, and g_z-invertible operators in terms of radial subspaces.
ER -