The g-Drazin inverses of anti-triangular block operator matrices. (English) Zbl 07764799
Summary: An element \(a\) in a Banach algebra \(\mathcal{A}\) has g-Drazin inverse if there exists \(b \in \mathcal{A}\) such that \(ab=ba\), \(b=bab\) and \(a-a^2 b \in \mathcal{A}^{qnil}\). In this paper we find new explicit representations of the g-Drazin inverse of the block operator matrix \(\begin{pmatrix} E & I \\ F & 0 \end{pmatrix}\). We thereby solve a wider kind of singular differential equations posed by S. L. Campbell [Linear Multilinear Algebra 14, 195–198 (1983; Zbl 0523.15007)].
MSC:
| 15A09 | Theory of matrix inversion and generalized inverses |
| 47A08 | Operator matrices |
| 46H05 | General theory of topological algebras |
Citations:
Zbl 0523.15007References:
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