On the Drazin inverse of the sum of two operators and its application to operator matrices

By Nieves Castro-González, Esther Dopazo, M. Martínez-Serrano. 2009

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Abstract Given two bounded linear operators F , G on a Banach space X such that G 2 F = G F 2 = 0 , we derive an explicit expression for the Drazin inverse of F + G . For this purpose, firstly, we obtain a formula for the resolvent of an auxiliary operator matrix in the form M = ( F I G F G ) . From the provided representation of ( F + G ) D several special cases are considered. In particular, we recover the case G F = 0 studied by Hartwig et al. [R.E. Hartwig, G. Wang, Y. Wei, Some additive results on Drazin inverse, Linear Algebra Appl. 322 (2001) 207–217] for matrices and by Djordjevic and Wei [D.S. Djordjevic, Y. Wei, Additive results for the generalized Drazin inverse, J. Aust. Math. Soc. 73 (1) (2002) 115–126] for operators. Finally, we apply our results to obtain representations for the Drazin inverse of operator matrices in the form M = ( A B C D ) which are extensions of some cases given in the literature.

Subfields: Algebra and Number Theory, Atomic and Molecular Physics, and Optics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/becf00c6faddd5c23e46d8d000c087e0d0605cca