Algebraic Direct Sums and Generalized Drazin Invertibility

Let $T$ be a bounded linear operator on a complex Banach space $X$. We prove that the algebraic decomposition $$X=\text{K}(T)\oplus \text{Hq}(T)$$ forces both summands to be closed and hence characterizes generalized Drazin invertibility. This improves the classical characterizations of Mbekhta~\resultcite{Theorem~1.6}{MbekhtaKato}, Schmoeger~\resultcite{Theorem~4}{Schmoeger}, and Dajić and Koliha~\resultcite{Theorem~6.7}{DajicKoliha}, by removing their closedness assumptions. We also show that $T$ is generalized Drazin invertible if and only if $$\dim (\text{K}(T)\cap \text{Hq}(T))<\mathrm{\infty }\text{and}\text{codim}(\text{K}(T)+\text{Hq}(T))<\mathrm{\infty }\mathrm{.}$$ For ordinary Drazin invertibility, we obtain analogous finite-defect characterizations involving the hyperkernel and the hyperrange. We also give a characterization using pointwise nilpotence on an invariant summand. Finally, we consider arbitrary invariant algebraic decompositions and give a counterexample when the inverse of the restriction to one summand is not assumed bounded.