A Generalization of the SMW Formula of Operator to the -Inverse Case
The classical Sherman-Morrison-Woodbury (for short SMW) formula is generalized to the -inverse case. Some sufficient conditions under which the SMW formula can be represented as are obtained.
There are numerous applications of the SMW formula in various fields (see [1–6, 12]). An excellent review by Hager  described some of the applications to statistics networks, structural analysis, asymptotic analysis, optimization, and partial differential equations. In this note, we consider the SMW formula in which the inverse is replaced by the -inverse. As we know, the inverse, the group inverse, the Moore-Penrose inverse, and the Drazin inverse all belong to the -inverse. Hence, the classical SMW formula is generalized.
Let and be complex Hilbert spaces. We denote the set of all bounded linear operators from to by and by when . For , let , let , and let be the adjoint, the range, and the null space of , respectively. If and both are invertible, and , then is invertible if and only if is invertible. In this case, The original SMW [1, 5, 6] formula (1) is only valid when is invertible. In particular, the SMW formula (1) implies that . An operator is called generalized invertible if there is an operator such that . The operator is not unique in general. In order to force its uniqueness, further conditions have to be imposed. The most likely convenient additional conditions are One also considers with some . Clearly, . Elements satisfying are called -inverse of , denoted by . Similarly, (, , and )-inverses are called group inverses, denoted by . (, , , and )-inverses are Moore-Penrose inverses (for short MP inverses), denoted by . It is well known that has the MP inverse if and only if is closed. And (, , and )-inverses are called Drazin inverses, denoted by (see ), where is the Drazin index of . In general, the -inverse of is not unique. It is clear that , are idempotents and if is invertible (see [8, 9]).
2. Main Results
The following lemmas are used to prove our main results.
Lemma 1. If and , then(i),(ii).
Lemma 2. Let . Then, , and .
Proof. Since , So, . Similarly, from there comes .
Theorem 3. Let , let , and let . Let also , and let . If then
For , let denote any kind of standard inverse , group inverse , MP inverse , and Drazin inverse, respectively. Since belongs to -inverse, we get the following corollary.
Corollary 4. Let , let , and let such that and exist. Let also , and let such that and exist. If then
The following is our second main result.
Theorem 5. Let , let , and let . Let also , and let . If any of the following items holds: then .
Proof. Define . We will prove that . (i)Since , , and , by Lemma 1, we have , , and . Hence, and .(ii)If , , and , then , , and . We have and
Corollary 6. Let , let , and let such that and exist. Let also , and let such that and exist. If then
Proof. Firstly, if denotes the standard inverse, then (15) holds automatically, and the result (SMW formula) follows immediately by the proof of Theorem 5.
Secondly, if denotes the MP inverse, by the proof of Theorem 5, satisfies and . Thus, , and . Moreover, we have and . By the definition of MP inverse, we know that .
Lastly, if denotes the Drazin inverse (resp., group inverse), by the proof of Theorem 5, satisfies . Moreover, , and is quasi-nilpotent (resp., for the group inverse case). So, by the definition of Drazin inverse, we have (resp. ).
3. Concluding Remark
In this note, we mainly extend the SMW formula to the form under some sufficient conditions. If , , , and in Theorem 3 are invertible, then (5) and (6) hold automatically. Hence, Theorem 3 generalizes the classical SMW formula. In [2, Theorem 2.1], when is MP-invertible and is invertible, the sufficient conditions under which the SMW formula can be represented as are given. It is obvious that Theorem 3 also generalizes [2, Theorem 2.3] greatly.
Under weaker assumptions than those used in the literature, our results are new and robust to the classical SMW formula even for the finite dimensional case. It is natural to ask if we can extend our results for the various inverses of in some weaker assumptions, which will be our future research topic.
The author thanks the anonymous referees for their careful reading, very detailed comments, and many constructive suggestions which greatly improved the presentation. She would like to thank Professors Hongke Du and Chunyuan Deng for their useful suggestions and comments. The paper is supported by the Natural Science Foundation of Zhanjiang Normal University under Grant QL1001.
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