The \(\sigma g\)-Drazin inverse and the generalized Mbekhta decomposition (Q2465107)

The authors define a generalization of the \(g\)-Drazin inverse of an element of a Banach algebra based on isolated spectral sets. Let \(\mathcal A\) be a unital Banach algebra. For any \(a\in {\mathcal A}\), let \(\text{Sp}(a)\) and \(\text{Res}(a)\) denote the spectrum and the resolvent set of \(a\) in the algebra \(\mathcal A\). A subset \(\sigma\) of \(\text{Sp}(a)\) is called an isolated spectral set of \(a\) if there exist disjoint open sets \(\Delta\supset \sigma\) and \(\Omega\supset \text{Sp}(a)\setminus \sigma\). To every isolated spectral set, there corresponds a spectral idempotent \(p\) obtained by the holomorphic functional calculus. By \(p'\) the spectral idempotent of \(\text{Sp}(a)\setminus\sigma\) is meant, i.e., \(p'=1-p\). An element \(a\in{\mathcal A}\) is called \(\sigma g\)-Drazin invertible if there exists an isolated spectral set \(\sigma\) of \(a\) such that \(0\in\text{Res}(a)\cup\sigma\). If \(a\) has this property, with \(p\) the spectral idempotent corresponding to \(\sigma\), then the inverse of \(ap'\) in the Banach algebra \(p'{\mathcal A}p'\) is called the \(\sigma g\)-Drazin inverse of \(a\) and is denoted by \(a^{\text{D},\sigma}\). The explicit formula for the \(\sigma g\)-Drazin inverse of \(a\) is given by the following Theorem. Let \(a\in{\mathcal A}\) be \(\sigma g\)-Drazin invertible with the spectral idempotent \(p\). Then, for any \(\xi\not\in\sigma\cup\{0\}\) and any \(\eta\not=0\), \[ a^{\text{D},\sigma}=(a-\xi p)^{-1}p'=(ap'-\eta p)^{-1}p'. \] Properties of this inverse are studied and some applications to differential equations are given. Also, generalized Mbekhta subspaces of a bounded linear operator on a Banach space are introduced. The generalized Mbekhta decomposition gives a characterization of circularly isolated spectral sets.