Independent Backward Preimages and Algebraic Radial Decompositions of Operators

Let T(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)ₛ(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.