For a matrix A ∈ Cn×n with index 1, the group inverse A. # is the unique solution of the and its unique solution is the Drazin inverse, or. {1k,2,5}–inverse, of A. PDF | The main theme of this paper can be described as a study of the Drazin inverse for bounded linear operators in a Banach space X when 0 is an isolated. 1. Introduction. The main theme of this paper can be described as a study of the Drazin inverse. In , Drazin [4] introduced a different kind of generalized.

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The paper is organized as follows.

Drazin inverse

From this, we can easily verify Therefore, ifthen. In recent years, the Drazin inverse of the sum of two matrices or operators has been extensively investigated under different conditions see, [ 5 — 15 ].

Since is nilpotent and is invertible, by Lemma 2. Journal of the Australian Mathematical Society A. Then, Thus By invegse above argument, we can get 3. We deduce the explicit expressions for and of two matrices and under the conditions and. Proof — If a is quasinilpotent, we can apply Theorem 3 and we obtain 15 for this particular case.

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It is obvious thatand Sinceis invertible and By 3. Before the theorem, let us recall that ifthen is invertible inversse Theorem 3. Ifthen and. New additive results for the g -Drazin inverse. Since andwe can writepartitioned conformably withby Lemma 2.

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Published online Feb 1. Then, for any positive integerwhere the binomial coefficient. Also, we give the upper bound of. Proceedings of the Royal Society of Edinburgh: So, by Lemma 2. First we start the following result which is proved in [ 8 ] for matrices, extended in [ 9 ] for a bounded linear operator and in [ 10 ] for arbitrary elements in a Banach algebra.

Moreover, if are nilpotent with andthen is nilpotent and its index is less than. Sign up or log in Sign up using Google. If with andthen Proof.

By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Post as a guest Name. Note that implies and. On the Drazin inverse of the sum of two operators and its application to operator matrices.

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A note on the Drazin inverse of an anti-triangular matrix. Some results on the generalized Drazin inverse of operator matrices. The index and the Drazin inverse of block triangular matrices. Ifthen is invertible, and therefore.

Drazin inverse – Wikipedia

The authors declare that there is no conflict of interests regarding the publication of this paper. TroyHaskin Ok, I’ll check it out. The next result is a generalization of [ 12Theorem 2. From 71 and 72 it follows ijverse Abstract We deduce the explicit expressions for and of two matrices and under the conditions and.