A Generalization of the SMW Formula of Operator 𝐴+𝑌𝐺𝑍^∗ to the {2}-Inverse Case

Abstract

The classical Sherman-Morrison-Woodbury (for short SMW) formula (𝐴+𝑌𝐺𝑍^∗)⁻¹=𝐴⁻¹−𝐴⁻¹𝑌(𝐺⁻¹+𝑍^∗𝐴⁻¹𝑌)⁻¹𝑍^∗𝐴⁻¹ is generalized to the {2}-inverse case. Some sufficient conditions under which the SMW formula can be represented as (𝐴+𝑌𝐺𝑍^∗)⁻=𝐴⁻−𝐴⁻𝑌(𝐺⁻+𝑍^∗𝐴⁻𝑌)⁻𝑍^∗𝐴⁻ are obtained.

1. Introduction

There are numerous applications of the SMW formula in various fields (see [1–6, 12]). An excellent review by Hager [3] 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 {2}-inverse. As we know, the inverse, the group inverse, the Moore-Penrose inverse, and the Drazin inverse all belong to the {2}-inverse. Hence, the classical SMW formula is generalized.

Let ℋ and 𝒦 be complex Hilbert spaces. We denote the set of all bounded linear operators from ℋ to 𝒦 by ℬ(ℋ,𝒦) and by ℬ(ℋ) when ℋ=𝒦. For 𝑇∈ℬ(ℋ,𝒦), let 𝑇^∗,  let ℛ(𝑇), and let 𝒩(𝑇) be the adjoint, the range, and the null space of 𝑇, respectively. If 𝐴∈ℬ(ℋ) and 𝐺∈ℬ(𝒦) both are invertible, and 𝑌,𝑍∈ℬ(𝒦,ℋ), then 𝐴+𝑌𝐺𝑍^∗ is invertible if and only if 𝐺⁻¹+𝑍^∗𝐴⁻¹𝑌 is invertible. In this case, 𝐴+𝑌𝐺𝑍^∗−1=𝐴⁻¹−𝐴⁻¹𝑌𝐺⁻¹+𝑍^∗𝐴⁻¹𝑌⁻¹𝑍^∗𝐴⁻¹. (1) The original SMW [1, 5, 6] formula (1) is only valid when 𝐴 is invertible. In particular, the SMW formula (1) implies that (𝐼−𝐴^∗𝐴)⁻¹=𝐼+𝐴^∗(𝐼−𝐴𝐴^∗)⁻¹𝐴. An operator 𝑇 is called generalized invertible if there is an operator 𝑆 such that (I)𝑇𝑆𝑇=𝑇. The operator 𝑆 is not unique in general. In order to force its uniqueness, further conditions have to be imposed. The most likely convenient additional conditions are (II)𝑆𝑇𝑆=𝑆,(III)(𝑇𝑆)^∗(=𝑇𝑆,IV)(𝑆𝑇)^∗=𝑆𝑇,(V)𝑇𝑆=𝑆𝑇. (2) One also considers (𝐼_𝑘)𝑇^𝑘𝑆𝑇=𝑇^𝑘 with some 𝑘∈𝐙⁺. Clearly, (I)=(I₁). Elements 𝑆∈ℬ(ℋ) satisfying (II) are called {2}-inverse of 𝑇, denoted by 𝑆=𝑇⁻. Similarly, (I, II, and V)-inverses are called group inverses, denoted by 𝑆=𝑇^#. (I, II, III, and IV)-inverses are Moore-Penrose inverses (for short MP inverses), denoted by 𝑆=𝑇⁺. It is well known that 𝑇 has the MP inverse if and only if ℛ(𝑇) is closed. And (I_𝑘, II, and V)-inverses are called Drazin inverses, denoted by 𝑆=𝑇^𝑑 (see [7]), where 𝑘 is the Drazin index of 𝑇. In general, the {2}-inverse of 𝑇 is not unique. It is clear that 𝑇𝑇⁻, 𝑇⁻𝑇 are idempotents and 𝑇⁺=𝑇^𝑑=𝑇^#=𝑇⁻=𝑇⁻¹ if 𝑇∈ℬ(ℋ) is invertible (see [8, 9]).

2. Main Results

The following lemmas are used to prove our main results.

Lemma 1. If 𝐴∈ℬ(ℋ) and 𝑃=𝑃²∈ℬ(ℋ), then(i)𝑃𝐴=𝐴⇔ℛ(𝐴)⊂ℛ(𝑃),(ii)𝐴𝑃=𝐴⇔𝒩(𝑃)⊂𝒩(𝐴).

It is well known that ℛ(𝐴)⊂ℛ(𝐵) if and only if there exists an operator 𝐶 such that 𝐴=𝐵𝐶 (see [10, 11]). So, we have the following result.

Lemma 2. Let 𝐴∈ℬ(ℋ,𝒦). Then, ℛ(𝐴⁻𝐴)=ℛ(𝐴⁻), and 𝒩(𝐴𝐴⁻)=𝒩(𝐴⁻).

Proof. Since 𝐴⁻=𝐴⁻𝐴𝐴⁻, ℛ(𝐴⁻)=ℛ(𝐴⁻𝐴𝐴⁻)⊂ℛ(𝐴⁻𝐴)⊂ℛ(𝐴⁻). (3) So, ℛ(𝐴⁻)=ℛ(𝐴⁻𝐴). Similarly, from 𝒩(𝐴⁻)⊂𝒩(𝐴𝐴⁻)⊂𝒩(𝐴⁻𝐴𝐴⁻)=𝒩(𝐴⁻), (4) there comes 𝒩(𝐴𝐴⁻)=𝒩(𝐴⁻).

Now, the first main result of this paper is given as follows, which generalizes the main results [2, Theorems 2.1–2.4] and [12, Theorem 1] with very short proof.

Theorem 3. Let 𝐴∈ℬ(ℋ), let 𝐺∈ℬ(𝒦), and let 𝑌,𝑍∈ℬ(𝒦,ℋ). Let also 𝐵=𝐴+𝑌𝐺𝑍^∗, and  let 𝑇=𝐺⁻+𝑍^∗𝐴⁻𝑌. If ℛ(𝐴⁻)⊂ℛ(𝐵⁻),𝒩(𝐴⁻)⊂𝒩(𝐵⁻),𝒩(𝐺⁻)⊂𝒩(𝑌),𝒩(𝑇⁻)⊂𝒩(𝐺), (5) then 𝐴+𝑌𝐺𝑍^∗−=𝐴⁻−𝐴⁻𝑌𝐺⁻+𝑍^∗𝐴⁻𝑌⁻𝑍^∗𝐴⁻. (6)

Proof. Let the conditions in (5) hold. By Lemma 2, we have ℛ(𝐴⁻)⊂ℛ(𝐵⁻)=ℛ(𝐵⁻𝐵),𝒩(𝐴𝐴⁻)=𝒩(𝐴⁻)⊂𝒩(𝐵⁻). (7) By Lemma 1, we get that 𝐵⁻𝐵𝐴⁻=𝐴⁻ and 𝐵⁻𝐴𝐴⁻=𝐵⁻. Hence, 𝐵⁻𝑌+𝐵⁻(𝐵−𝐴)𝐴⁻𝑌=𝐴⁻𝑌. Similarly, from 𝒩(𝐺⁻)⊂𝒩(𝑌), we get that 𝑌𝐺𝐺⁻=𝑌 and 𝐵⁻𝑌𝐺𝑇=𝐵⁻𝑌𝐺𝐺⁻+𝐵⁻𝑌𝐺𝑍^∗𝐴⁻𝑌=𝐵⁻𝑌+𝐵⁻(𝐵−𝐴)𝐴⁻𝑌=𝐴⁻𝑌. (8) So, the condition 𝒩(𝑇⁻)⊂𝒩(𝐺) implies that 𝐵⁻𝑌𝐺=𝐵⁻𝑌𝐺𝑇𝑇⁻=𝐴⁻𝑌𝑇⁻. From 𝐵=𝐴+𝑌𝐺𝑍^∗, we deduce that 𝐵⁻𝐵𝐴⁻=𝐵⁻𝐴𝐴⁻+𝐵⁻𝑌𝐺𝑍^∗𝐴⁻. Hence, 𝐵⁻=𝐵⁻𝐴𝐴⁻=𝐵⁻𝐵𝐴⁻−𝐵⁻𝑌𝐺𝑍^∗𝐴⁻=𝐴⁻−𝐴⁻𝑌𝑇⁻𝑍^∗𝐴⁻. (9)

For 𝑇∈ℬ(ℋ), let 𝑇^⨀ denote any kind of standard inverse 𝑇⁻¹, group inverse 𝑇^#, MP inverse 𝑇⁺, and Drazin inverse, respectively. Since 𝑇^⨀ belongs to {2}-inverse, we get the following corollary.

Corollary 4. Let 𝐴∈ℬ(ℋ), let 𝐺∈ℬ(𝒦), and let 𝑌,𝑍∈ℬ(𝒦,ℋ) such that 𝐴^⨀ and 𝐺^⨀ exist. Let also 𝐵=𝐴+𝑌𝐺𝑍^∗, and let 𝑇=𝐺^⨀+𝑍^∗𝐴^⨀𝑌 such that 𝐵^⨀ and 𝑇^⨀ exist. If ℛ𝐴^⨀⊂ℛ𝐵^⨀,𝒩𝐴^⨀⊂𝒩𝐵^⨀,𝒩𝐺^⨀⊂𝒩(𝑌),𝒩𝑇^⨀⊂𝒩(𝐺), (10) then 𝐴+𝑌𝐺𝑍^∗⨀=𝐴^⨀−𝐴^⨀𝑌𝐺^⨀+𝑍^∗𝐴^⨀𝑌^⨀𝑍^∗𝐴^⨀. (11)

The following is our second main result.

Theorem 5. Let 𝐴∈ℬ(ℋ), let 𝐺∈ℬ(𝒦), and let 𝑌,𝑍∈ℬ(𝒦,ℋ). Let also 𝐵=𝐴+𝑌𝐺𝑍^∗, and let 𝑇=𝐺⁻+𝑍^∗𝐴⁻𝑌. If any of the following items holds: (𝑖)𝒩(𝐴⁻𝐴)⊂𝒩𝑍^∗,ℛ𝑍^∗⊂ℛ(𝐺⁻),𝒩(𝑇⁻𝑇)(𝑌),()𝒩(𝐺⊂𝒩𝑖𝑖⁻)⊂𝒩(𝑌),ℛ𝑍^∗⊂ℛ(𝑇𝑇⁻),ℛ(𝑌)⊂ℛ(𝐴𝐴⁻), (12) then (𝐴+𝑌𝐺𝑍^∗)⁻=𝐴⁻−𝐴⁻𝑌(𝐺⁻+𝑍^∗𝐴⁻𝑌)⁻𝑍^∗𝐴⁻.

Proof. Define 𝑋=𝐴⁻−𝐴⁻𝑌𝑇⁻𝑍^∗𝐴⁻. We will prove that 𝑋𝐵𝑋=𝑋. (i)Since 𝒩(𝐴⁻𝐴)⊂𝒩(𝑍^∗), 𝒩(𝑇⁻𝑇)⊂𝒩(𝑌), and ℛ(𝑍^∗)⊂ℛ(𝐺⁻), by Lemma 1, we have 𝑍^∗(𝐼−𝐴⁻𝐴)=0, 𝑌(𝐼−𝑇⁻𝑇)=0, and (𝐼−𝐺⁻𝐺)𝑍^∗=0. Hence, 𝑋𝐵=𝐴⁻−𝐴⁻𝑌𝑇⁻𝑍^∗𝐴⁻𝐴+𝑌𝐺𝑍^∗=𝐴⁻𝐴+𝐴⁻𝑌𝐺𝑍^∗−𝐴⁻𝑌𝑇⁻𝑍^∗𝐴⁻𝐴−𝐴⁻𝑌𝑇⁻𝑍^∗𝐴⁻𝑌𝐺𝑍^∗=𝐴⁻𝐴+𝐴⁻𝑌𝐺𝑍^∗−𝐴⁻𝑌𝑇⁻𝑍^∗−𝐴⁻𝑌𝑇⁻(𝑇−𝐺⁻)𝐺𝑍^∗=𝐴⁻𝐴+𝐴⁻𝑌(𝐼−𝑇⁻𝑇)𝐺𝑍^∗−𝐴⁻𝑌𝑇⁻(𝐼−𝐺⁻𝐺)𝑍^∗=𝐴⁻𝐴, (13) and 𝑋𝐵𝑋=𝐴⁻𝐴𝑋=𝑋.(ii)If 𝒩(𝐺⁻)⊂𝒩(𝑌), ℛ(𝑍^∗)⊂ℛ(𝑇𝑇⁻), and ℛ(𝑌)⊂ℛ(𝐴𝐴⁻), then 𝑌(𝐼−𝐺𝐺⁻)=0, (𝐼−𝑇𝑇⁻)𝑍^∗=0, and (𝐼−𝐴𝐴⁻)𝑌=0. We have 𝐵𝑋=𝐴+𝑌𝐺𝑍^∗𝐴⁻−𝐴⁻𝑌𝑇⁻𝑍^∗𝐴⁻=𝐴𝐴⁻+𝑌𝐺𝑍^∗𝐴⁻−𝐴𝐴⁻𝑌𝑇⁻𝑍^∗𝐴⁻−𝑌𝐺𝑍^∗𝐴⁻𝑌𝑇⁻𝑍^∗𝐴⁻=𝐴𝐴⁻+𝑌𝐺𝑍^∗𝐴⁻−𝑌𝑇⁻𝑍^∗𝐴⁻−𝑌𝐺(𝑇−𝐺⁻)𝑇⁻𝑍^∗𝐴⁻=𝐴𝐴⁻+𝑌𝐺(𝐼−𝑇𝑇⁻)𝑍^∗𝐴⁻−𝑌(𝐼−𝐺𝐺⁻)𝑇⁻𝑍^∗𝐴⁻=𝐴𝐴⁻ (14) and 𝑋𝐵𝑋=𝑋𝐴𝐴⁻=𝑋.

From Theorem 5, we get the following corollary which generalizes some recent results (e.g., see [2, Theorem 2.1] and [12, Theorem 1]).

Corollary 6. Let 𝐴∈ℬ(ℋ), let 𝐺∈ℬ(𝒦), and let 𝑌,𝑍∈ℬ(𝒦,ℋ) such that 𝐴^⨀ and 𝐺^⨀ exist. Let also 𝐵=𝐴+𝑌𝐺𝑍^∗, and let 𝑇=𝐺^⨀+𝑍^∗𝐴^⨀𝑌 such that 𝐵^⨀ and 𝑇^⨀ exist. If 𝒩𝐴^⨀𝐴⊂𝒩𝑍^∗,ℛ𝑍^∗⊂ℛ𝐺^⨀,𝒩𝑇^⨀𝑇⊂𝒩(𝑌),𝒩𝐺^⨀(𝑌),ℛ𝑍⊂𝒩_∗⊂ℛ𝑇𝑇^⨀,ℛ(𝑌)⊂ℛ𝐴𝐴^⨀, (15) then 𝐴+𝑌𝐺𝑍^∗⨀=𝐴^⨀−𝐴^⨀𝑌𝐺^⨀+𝑍^∗𝐴^⨀𝑌^⨀𝑍^∗𝐴^⨀. (16)

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, 𝑋=𝐴⁺−𝐴⁺𝑌𝑇⁺𝑍^∗𝐴⁺ satisfies 𝑋𝐵=𝐴⁺𝐴 and 𝐵𝑋=𝐴𝐴⁺. Thus, (𝑋𝐵)^∗=𝑋𝐵, and (𝐵𝑋)^∗=𝐵𝑋. Moreover, we have 𝐵𝑋𝐵=𝐴𝐴⁺(𝐴+𝑌𝐺𝑍^∗)=𝐴+𝐴𝐴⁺𝑌𝐺𝑍^∗=𝐵 and 𝑋𝐵𝑋=𝐴⁺𝐴(𝐴⁺−𝐴⁺𝑌𝑇⁺𝑍^∗𝐴⁺)=𝑋. By the definition of MP inverse, we know that (𝐴+𝑌𝐺𝑍^∗)⁺=𝐴⁺−𝐴⁺𝑌(𝐺⁺+𝑍^∗𝐴⁺𝑌)⁺𝑍^∗𝐴⁺.
Lastly, if ⨀ denotes the Drazin inverse (resp., group inverse), by the proof of Theorem 5, 𝑋=𝐴^𝑑−𝐴^𝑑𝑌𝑇^𝑑𝑍^∗𝐴^𝑑 satisfies 𝑋𝐵=𝐵𝑋=𝐴𝐴^𝑑=𝐴^𝑑𝐴. Moreover, 𝑋𝐵𝑋=𝐴^𝑑𝐴𝑋=𝐴^𝑑𝐴(𝐴^𝑑−𝐴^𝑑𝑌𝑇^𝑑𝑍^∗𝐴^𝑑)=𝑋, and 𝐵−𝐵²𝑋=𝐵(𝐼−𝐵𝑋)=(𝐴+𝑌𝐺𝑍^∗)(𝐼−𝐴𝐴^𝑑)=𝐴(𝐼−𝐴𝐴^𝑑) is quasi-nilpotent (resp.,  𝐵−𝐵²𝑋=0 for the group inverse case). So, by the definition of Drazin inverse, we have (𝐴+𝑌𝐺𝑍^∗)^𝑑=𝐴^𝑑−𝐴^𝑑𝑌𝑇^𝑑𝑍^∗𝐴^𝑑 (resp. (𝐴+𝑌𝐺𝑍^∗)^#=𝐴^#−𝐴^#𝑌𝑇^#𝑍^∗𝐴^#).

3. Concluding Remark

In this note, we mainly extend the SMW formula to the form (𝐴+𝑌𝐺𝑍^∗)⁻=𝐴⁻−𝐴⁻𝑌(𝐺⁻+𝑍^∗𝐴⁻𝑌)⁻𝑍^∗𝐴⁻ under some sufficient conditions. If 𝐴, 𝐺, 𝑇, and 𝐵 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 𝐴 is MP-invertible and 𝐺⁻¹+𝑍^∗𝐴⁺𝑌 is invertible, the sufficient conditions under which the SMW formula can be represented as 𝐴+𝑌𝐺𝑍^∗+=𝐴⁺−𝐴⁺𝑌𝐺⁻¹+𝑍^∗𝐴⁻𝑌⁻¹𝑍^∗𝐴⁺ (17) 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 𝐴+𝑌𝐺𝑍^∗ in some weaker assumptions, which will be our future research topic.

Acknowledgments

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.