Algebraic Direct Sums and Generalized Drazin Invertibility

Abstract

Let $T$ be a bounded linear operator on a complex Banach space $X$. We prove that the algebraic decomposition \[ X=\K(T)\oplus \Hq(T) \] 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 \[ \dim\bigl(\K(T)\cap \Hq(T)\bigr)<\infty \quad\text{and}\quad \codim\bigl(\K(T)+\Hq(T)\bigr)<\infty. \] 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.