International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 846165 | https://doi.org/10.5402/2011/846165

Chun Yuan Deng, "On the Idempotent Solutions of a Kind of Operator Equations", International Scholarly Research Notices, vol. 2011, Article ID 846165, 11 pages, 2011. https://doi.org/10.5402/2011/846165

On the Idempotent Solutions of a Kind of Operator Equations

Academic Editor: F. Tadeo
Received11 Mar 2011
Accepted19 Apr 2011
Published08 Jun 2011

Abstract

This paper provides some relations between the idempotent operators and the solutions to operator equations 𝐴𝐵𝐴=𝐴2 and 𝐵𝐴𝐵=𝐵2.

1. Introduction

Let ℋ be a complex Hilbert space. Denote by ℬ(ℋ) the Banach algebra of all bounded linear operators on ℋ. For 𝐴,𝐵∈ℬ(ℋ), if 𝐴 and 𝐵 satisfy the relations𝐴𝐵𝐴=𝐴2,𝐵𝐴𝐵=𝐵2,(1.1) we say the pair of (𝐴,𝐵) is the solution to (1.1). In [1], Vidav has investigated the self-adjoint solutions to (1.1) and showed that the pair of (𝐴,𝐵) is self-adjoint solution to (1.1) if and only if there exists unique idempotent operator 𝑃 such that 𝐴=𝑃𝑃∗ and 𝐵=𝑃∗𝑃. In [2], Rakočević gave another proof of this result by using some properties of generalized inverses. In [3], Schmoeger generalized the Vidav's result concerning (1.1) by using some properties of Drazin inverses. The aim of this paper is to investigate some connections between idempotent operators and the solutions to (1.1). We prove main results as follows.(1)𝐴 and 𝐵 are idempotent solution to (1.1) if and only if there exist idempotent operators 𝑃 and 𝑄 satisfying (1.1) such that 𝐴=𝑃𝑄 and 𝐵=𝑄𝑃.(2)If 𝐴 is generalized Drazin invertible such that 𝐴𝜋𝐵(𝐼−𝐴𝜋)=0. Then 𝐴 and 𝐵 satisfy (1.1) if and only if 𝐴=𝑃1+𝑁1 and 𝐵=𝑃2+𝑁2, where 𝑁1 and 𝑁2 are arbitrary quasinilpotent elements satisfying (1.1), 𝑃1 and 𝑃2 are arbitrary idempotent elements satisfying ℛ(𝑃1)=ℛ(𝑃2) and 𝑃𝑖⟂𝑐𝑁𝑗, 𝑖,𝑗=1,2.

Before proving the main results in this paper, let us introduce some notations and terminology which are used in the later. For 𝑇∈ℬ(ℋ), we denote by ℛ(𝑇), 𝒩(𝑇), ğœŽğ‘(𝑇) and ğœŽ(𝑇) the range, the null space, the point spectrum, and the spectrum of 𝑇, respectively. An operator 𝑃∈ℬ(ℋ) is said to be idempotent if 𝑃2=𝑃. 𝑃 is called an orthogonal projection if 𝑃=𝑃2=𝑃∗, where 𝑃∗ denotes the adjoint of 𝑃. An operator 𝐴∈ℬ(ℋ) is unitary if 𝐴𝐴∗=𝐴∗𝐴=𝐼. 𝐴 is positive if (𝐴𝑥,𝑥)≥0 for all 𝑥∈ℋ and its unique positive square root is denoted by 𝐴1/2. For a closed subspace 𝒦 of ℋ,𝑇|𝒦 denotes the restriction of 𝑇 on 𝒦 and 𝑃𝒦 denotes the orthogonal projection onto 𝒦. The generalized Drazin inverse (see [4, 5]) is the element 𝑇𝑑∈ℬ(ℋ) such that𝑇𝑇𝑑=𝑇𝑑𝑇,𝑇𝑑𝑇𝑇𝑑=𝑇𝑑,𝑇−𝑇2𝑇𝑑isquasinilpotent.(1.2) It is clear 𝑇𝑑=𝑇−1 if 𝑇∈ℬ(ℋ) is invertible. If 𝑇 is generalized Drazin invertible, then the spectral idempotent 𝑇𝜋 of 𝑇 corresponding to {0} is given by 𝑇𝜋=𝐼−𝑇𝑇𝑑. The operator matrix form of 𝑇 with respect to the space decomposition ℋ=𝒩(𝑇𝜋)⊕ℛ(𝑇𝜋) is given by 𝑇=𝑇1⊕𝑇2, where 𝑇1 is invertible and 𝑇2 is quasinilpotent.

2. Some Lemmas

To prove the main results, some lemmas are needed.

Lemma 2.1 (see [6, Lemma 1.1]). Let 𝑃 be an idempotent in ℬ(ℋ). Then there exists an invertible operator 𝑆∈ℬ(ℋ) such that SPS−1 is an orthogonal projection.

Lemma 2.2 (see [7, Theorem 2.1]). Let 𝑃,𝑄∈ℬ(ℋ) with 𝑃=𝑃2 and 𝑄=𝑄2=𝑄∗. If ℛ(𝑃)=ℛ(𝑄), then 𝑃+𝑃∗−𝐼 is always invertible and 𝑄=𝑃𝑃+𝑃∗−𝐼−1=𝑃+𝑃∗−𝐼−1𝑃∗.(2.1)

Lemma 2.3 (see [8, Remark 1.2.1]). Let 𝐴,𝐵∈ℬ(ℋ). Then ğœŽ(𝐴𝐵)⧵{0}=ğœŽ(𝐵𝐴)⧵{0}.

Lemma 2.4 (see [9, 10]). Let 𝐴∈ℬ(𝐻) have the matrix form 𝐴=𝐴11𝐴12𝐴21𝐴22. Then 𝐴≥0 if and only if 𝐴𝑖𝑖≥0, 𝑖=1,2, 𝐴21=𝐴∗12 and there exists a contraction operator 𝐷 such that 𝐴12=𝐴1/211𝐷𝐴1/222.

Lemma 2.5. Let 𝐴,𝐵∈ℬ(ℋ) with 𝐴𝐵𝐴=𝐴2 and 𝐵𝐴𝐵=𝐵2. Then (𝐴𝐵)𝑘=𝐴𝑘𝐵=𝐴𝐵𝑘,𝐴𝑘𝐵𝑙=𝐴𝑘+𝑙−1𝐵=𝐴𝐵𝑘+𝑙−1(2.2) for all nonnegative integer 𝑘,𝑙≥1.

Proof. The conditions 𝐴𝐵𝐴=𝐴2 and 𝐵𝐴𝐵=𝐵2 imply that (𝐴𝐵)2=𝐴𝐵𝐴𝐵=𝐴2𝐵=𝐴𝐵2. Now suppose (𝐴𝐵)𝑘=𝐴𝑘𝐵=𝐴𝐵𝑘 holds for nonnegative integer 2≤𝑘≤𝑚. Then, for 𝑘=𝑚+1, we have (𝐴𝐵)𝑚+1=(𝐴𝐵)2(𝐴𝐵)𝑚−1=𝐴2𝐵(𝐴𝐵)𝑚−1=𝐴(𝐴𝐵)𝑚=𝐴𝑚+1𝐵=𝐴𝐴𝑚𝐵=𝐴2𝐵𝑚=𝐴𝐵2𝐵𝑚−1=𝐴𝐵𝑚+1.(2.3) Hence (𝐴𝐵)𝑘=𝐴𝑘𝐵=𝐴𝐵𝑘 and 𝐴𝑘𝐵𝑙=𝐴𝑘−1𝐴𝐵𝑙=𝐴𝑘−1𝐴𝑙𝐵=𝐴𝑘+𝑙−1𝐵=𝐴𝐵𝑘+𝑙−1 for all nonnegative integer 𝑘,𝑙≥1.

An element 𝑇∈ℬ(ℋ) whose spectrum ğœŽ(𝑇) consists of the set {0} is said to be quasi-nilpotent [8]. It is clear that 𝑇 is quasi-nilpotent if and only if the spectral radius 𝛾(𝑇)=sup{|𝜆|âˆ¶ğœ†âˆˆğœŽ(𝑇)}=0. In particular, if there exists a positive integer 𝑚 such that 𝐴𝑚=0, then 𝐴 is 𝑚-nilpotent element. For the quasi-nilpotent operator, we have the following results.

Lemma 2.6. Let 𝐴 and 𝐵 satisfy (1.1). Then 𝐴 is quasinilpotent if and only if 𝐵 is quasinilpotent. In particular, 𝐴 is nilpotent if and only if 𝐵 is nilpotent; if 𝐴 is quasinilpotent and 𝐴𝐵=𝐵𝐴, then 𝐴2=𝐵2=0.

Proof. Because ğœŽ(𝐴2)∪{0}=ğœŽ(𝐴𝐵𝐴)∪{0}=ğœŽ(𝐴2𝐵)∪{0}=ğœŽ(𝐴𝐵2)∪{0}=ğœŽ(𝐵𝐴𝐵)∪{0}=ğœŽ(𝐵2)∪{0}, it follows that 𝐴 is quasinilpotent if and only if 𝐵 is quasinilpotent.

By Lemma 2.5, if there is a nonnegative integer 𝑚≥1 such that 𝐴𝑚=0, then 𝐵𝑚+1=𝐵𝑚−1𝐵𝐴𝐵=𝐵𝐴𝑚𝐵=0.(2.4) Similarly we can show that 𝐴𝑛+1=0 if 𝐵𝑛=0. Hence 𝐴 is nilpotent if and only if 𝐵 is nilpotent.

If 𝐴 is quasinilpotent, then 𝐵 is quasinilpotent, so 𝐼−𝐴 and 𝐼−𝐵 are invertible. From the condition 𝐴𝐵=𝐵𝐴, we obtain 𝐴2(𝐼−𝐵)=𝐴2−𝐴𝐵𝐴=0,𝐵2(𝐼−𝐵)=𝐵2−𝐵𝐴𝐵=0. It follows 𝐴2=0 and 𝐵2=0.

Lemma 2.7. Let 𝐴 and 𝐵 satisfy (1.1). Then for every integer 𝑘≥1, ğœŽî€·ğ´2𝐵=ğœŽ2𝐴,ğœŽğ‘˜ğµî€¸î€·ğµ=ğœŽğ‘˜ğ´î€¸.(2.5)

Proof. Since 𝐴2𝐵=𝐴𝐵2 by Lemma 2.5, ğœŽî€·ğ´2𝐵𝐴∪{0}=ğœŽ(𝐴𝐵𝐴)∪{0}=ğœŽ2î€¸ğœŽî€·âˆª{0},𝐴𝐵2𝐵∪{0}=ğœŽ(𝐵𝐴𝐵)∪{0}=ğœŽ2∪{0}.(2.6) Note that 𝐴 is invertible if and only if 𝐵 is invertible. We get ğœŽ(𝐴2)=ğœŽ(𝐵2). Next, if 0âˆ‰ğœŽ(𝐴𝐵), then from (𝐴𝐵)2=𝐴2𝐵 we obtain 𝐴𝐵=𝐴, that is, 𝐴 is invertible. It follows that 𝐴=𝐵=𝐼 because 𝐴𝐵𝐴=𝐴2 and 𝐵𝐴𝐵=𝐵2, so ğœŽ(𝐴𝐵)=ğœŽ(𝐵𝐴). Now, ğœŽî€·ğ´ğ‘˜ğµî€¸î€·=ğœŽ(𝐴𝐵)𝑘=𝜆𝑘,ğœŽî€·ğµâˆ¶ğœ†âˆˆğœŽ(𝐴𝐵)𝑘𝐴=ğœŽ(𝐵𝐴)𝑘=𝜇𝑘.âˆ¶ğœ‡âˆˆğœŽ(𝐵𝐴)(2.7) Hence, ğœŽ(𝐴𝑘𝐵)=ğœŽ(𝐵𝑘𝐴) for every integer 𝑘≥1.

3. Idempotent Solutions

In this section, we will show that the solutions to (1.1) have a closed connection with the idempotent operators. Our main results are as follows.

Theorem 3.1. The following assertions are equivalent. (a)𝐴 and 𝐵 are idempotent solution to (1.1).(b)There exist idempotent operators 𝑃 and 𝑄 satisfying (1.1) such that 𝐴=𝑃𝑄,𝐵=𝑄𝑃.(3.1)

Proof. Clearly, we only needs prove that (a) implies (b). Since 𝐴 and 𝐵 are idempotent operators, 𝐴𝐵𝐴=𝐴2 and 𝐵𝐴𝐵=𝐵2, without loss of generality, we can assume that one of 𝐴 and 𝐵 is orthogonal projection by Lemma 2.1. For example, assume that 𝐵 is an orthogonal projection. From 𝐴𝐵𝐴=𝐴, we obtain 𝒩(𝐵|ℛ(𝐴))=0. Since 𝐵 is an orthogonal projection and 𝐵𝐴𝐵=𝐵, we have 𝐵𝐴∗𝐵=𝐵 and 𝒩(𝐼|ℛ(𝐴)⟂−𝐵|ℛ(𝐴)⟂)=0. By Lemma 2.4, 𝐴 and 𝐵 can be written in the forms of âŽ›âŽœâŽœâŽœâŽœâŽœâŽğ´=𝐼0𝑃13𝑃14𝐼𝑃23𝑃2400âŽžâŽŸâŽŸâŽŸâŽŸâŽŸâŽ âŽ›âŽœâŽœâŽœâŽœâŽœâŽğ¼ğ‘„,𝐵=1𝑄11/2𝐷𝑄21/2𝑄21/2𝐷∗𝑄11/2𝑄20⎞⎟⎟⎟⎟⎟⎠,(3.2) with respect to the space decomposition ∑ℋ=4𝑖=1⊕ℋ𝑖, respectively, where ℋ1=𝒩(𝐼ℛ(𝐴)−𝐵|ℛ(𝐴)), ℋ2=ℛ(𝐴)⊖ℋ1, ℋ4=𝒩(𝐵|ℛ(𝐴)⟂), ℋ3=ℛ(𝐴)⟂⊖ℋ4 and the entries omitted are zero. It is easy to see that 𝑄𝑖 as operators on ℋ1+𝑖,𝑖=1,2, are injective positive contractions, and 𝐷 is a contraction from ℋ3 into ℋ2 by Lemma 2.4. Since 𝐵 is an orthogonal projection, 𝑄1𝑄11/2𝐷𝑄21/2𝑄21/2𝐷∗𝑄11/2𝑄22=𝑄1𝑄11/2𝐷𝑄21/2𝑄21/2𝐷∗𝑄11/2𝑄2,(3.3) that is, 𝑄21+𝑄11/2𝐷𝑄2𝐷∗𝑄11/2𝑄13/2𝐷𝑄21/2+𝑄11/2𝐷𝑄23/2𝑄21/2𝐷∗𝑄13/2+𝑄23/2𝐷∗𝑄11/2𝑄22+𝑄21/2𝐷∗𝑄1𝐷𝑄21/2=𝑄1𝑄11/2𝐷𝑄21/2𝑄21/2𝐷∗𝑄11/2𝑄2.(3.4) Comparing both sides of the above equation and observing that self-adjoint operators 𝑄𝑖, 𝐼−𝑄𝑖, 𝑖=1,2 are injective, by a straightforward computation we obtain 𝑄2=𝐷∗𝐼−𝑄1𝐷,𝐷𝐷∗=𝐼,𝐷∗𝐷=𝐼.(3.5) Hence âŽ›âŽœâŽœâŽğ‘„ğµ=𝐼⊕1𝑄11/2𝐼−𝑄11/2𝐷𝐷∗𝐼−𝑄11/2𝑄11/2𝐷∗𝐼−𝑄1î€¸ğ·âŽžâŽŸâŽŸâŽ âŠ•0,(3.6) where 0 and 1 are not in ğœŽğ‘(𝑄1), 𝐷 is unitary from ℋ3 onto ℋ2 (see [11] and Lemma 1 in [12]). Denote by 𝐴𝐵𝐴=(𝑇𝑖𝑗)1≤𝑖,𝑗≤4. A direct computation shows that 𝑇12=𝑃13𝐷∗𝑄11/2𝐼−𝑄11/2,𝑇22=𝑄1+𝑃23𝐷∗𝑄11/2𝐼−𝑄11/2,(3.7) and 𝐴𝐵𝐴=𝐴 if and only if 𝑇12=0 and 𝑇22=𝐼. Since 𝑄1 and 𝐼−𝑄1 injective self-adjoint operators, we obtain 𝑃13=0 and 𝑃23𝐷∗𝑄11/2=(𝐼−𝑄1)1/2. Moreover, we can show 𝐵𝐴𝐵=𝐵 when 𝑃13=0 and 𝑃23𝐷∗𝑄11/2=(𝐼−𝑄1)1/2. Hence, âŽ›âŽœâŽœâŽœâŽœâŽœâŽğ´=𝐼00𝑃14𝐼𝑃23𝑃2400âŽžâŽŸâŽŸâŽŸâŽŸâŽŸâŽ âŽ›âŽœâŽœâŽœâŽœâŽœâŽğ¼ğ‘„,𝐵=1𝑄11/2𝐼−𝑄11/2𝐷𝐷∗𝐼−𝑄11/2𝑄11/2𝐷∗𝐼−𝑄1𝐷0⎞⎟⎟⎟⎟⎟⎠,(3.8) where 𝑄1 is a contraction on ℋ2, 0 and 1 are not in ğœŽğ‘(𝑄1), 𝐷 is unitary from ℋ3 onto ℋ2, 𝑃𝑖4∈ℬ(ℋ4,ℋ𝑖),𝑖=1,2 are arbitrary, 𝑃23∈ℬ(ℋ3,ℋ2) and 𝑃23𝐷∗𝑄11/2=(𝐼−𝑄1)1/2. Let 𝑃=𝐼⊕𝐼𝑃23⎛⎜⎜⎜⎜⎜⎝00⊕0,𝑄=𝐼𝑃14𝑄1𝑄11/2𝐼−𝑄11/2𝐷𝑄1𝑃24𝐷∗𝐼−𝑄11/2𝑄11/2𝐷∗𝐼−𝑄1𝐷𝐷∗𝐼−𝑄11/2𝑄11/2𝑃240⎞⎟⎟⎟⎟⎟⎠.(3.9) Then we can deduce that idempotent operators 𝑃 and 𝑄 satisfy (1.1), 𝑃𝑄=𝐴 and 𝑄𝑃=𝐵.

Theorem 3.1 shows that the arbitrary pair of idempotent solution (𝐴,𝐵) can be written as 𝐴=𝑃𝑄, 𝐵=𝑄𝑃 with idempotent operators 𝑃 and 𝑄 satisfying (1.1). Next, we discuss the uniqueness of the idempotent solution to (1.1).

Theorem 3.2. Let 𝐵 be given idempotent. Then (1.1) has unique idempotent solution 𝐴 if and only if 𝒩(𝐵|ℛ(𝐴)⟂)=0. In this case, 𝐴,𝐵 satisfy 𝐴𝐵=𝐴 and 𝐵𝐴=𝐵.

Proof. Suppose that the pair (𝐴,𝐵) being the idempotent solution to (1.1). By the proof of Theorem 3.1, if ℋ4=𝒩(𝐵|ℛ(𝐴)⟂)≠0, the idempotent solution 𝐴 is not unique because 𝑃14 and 𝑃24 are arbitrary elements; if ℋ4=𝒩(𝐵|ℛ(𝐴)⟂)=0, then 𝐴 and 𝐵 have the form âŽ›âŽœâŽœâŽœâŽğ´=𝐼00𝐼𝑃230âŽžâŽŸâŽŸâŽŸâŽ âŽ›âŽœâŽœâŽœâŽœâŽğ¼ğ‘„,𝐵=1𝑄11/2𝐼−𝑄11/2𝐷𝐷∗𝐼−𝑄11/2𝑄11/2𝐷∗𝐼−𝑄1î€¸ğ·âŽžâŽŸâŽŸâŽŸâŽŸâŽ .(3.10) Since 𝑄1=𝑄∗1 is injection, 𝐷 is unitary and 𝑃23𝐷∗𝑄11/2=(𝐼−𝑄1)1/2, so 𝑄11/2𝐷𝑃∗23=(𝐼−𝑄1)1/2 and 𝑃∗23=𝐷∗𝑄1−1/2(𝐼−𝑄1)1/2. Hence, we obtain that 𝑃23 is uniquely determined and 𝑃23=𝑄1−1/2(𝐼−𝑄1)1/2𝐷. Therefore the idempotent solution 𝐴 is unique and 𝐴𝐵=𝐴 and 𝐵𝐴=𝐵.

The following result was first given by Vidav [1]. We give an alternative short proof.

Theorem 3.3 (see [1, Theorem 2]). The following assertions are equivalent. (a)𝐴 and 𝐵 are self-adjoint solution to (1.1).(b)There is an idempotent operator 𝑃 such that 𝐴=𝑃𝑃∗,𝐵=𝑃∗𝑃.(3.11)

Proof. (b) implies (a) is clear. Now, suppose that (a) holds. From 𝐴(𝐵−𝐼)2𝐴=𝐴𝐵2𝐴−2𝐴𝐵𝐴+𝐴2=𝐴2𝐵𝐴−𝐴2=𝐴3−𝐴2,(3.12) we have 𝐴3−𝐴2=𝐴(𝐵−𝐼)(𝐴(𝐵−𝐼))∗≥0, so ğœŽ(𝐴3−𝐴2)⊂[0,∞). The spectral mapping theorem gives 𝜆3−𝜆2≥0,âˆ€ğœ†âˆˆğœŽ(𝐴)⧵{0}.(3.13) Thus, for âˆ€ğœ†âˆˆğœŽ(𝐴)⧵{0}, we have 𝜆≥1 and therefore 𝐴≥0. ℛ(𝐴) is closed since 0 is not the accumulation point of ğœŽ(𝐴). Hence 𝐴 has the matrix form 𝐴=𝐴1⊕0 according to the space decomposition ℋ=ℛ(𝐴)⊕ℛ⟂(𝐴), where 𝐴1 is invertible. Similarly, we can derive that 𝐵≥0. By Lemma 2.4, 𝐵 can be written as 𝐵=𝐵1𝐵2𝐵∗2𝐵4 with 𝐵1≥0 and 𝐵4≥0. From 𝐴𝐵𝐴=𝐴2, we have 𝐵1=𝐼. From 𝐵𝐴𝐵=𝐵2, we have 𝐼+𝐵2𝐵∗2=𝐴1,𝐵∗2𝐵2+𝐵24=𝐵∗2𝐴1𝐵2⟹𝐵24=𝐵∗2𝐵22⟹𝐵4=𝐵∗2𝐵2,(3.14) since the square root of a positive operator is unique. Define 𝑃=𝐼𝐵200. Then 𝑃2=𝑃, 𝑃𝑃∗=𝐼+𝐵2𝐵∗2000=𝐴,𝑃∗𝑃=𝐼𝐵2𝐵∗2𝐵∗2𝐵2=𝐵.(3.15)

The next characterizations of the solutions to (1.1) are clear.

Corollary 3.4. (a) For arbitrary idempotent operators 𝑃 and 𝑄, 𝐴=𝑃𝑄, 𝐵=𝑄𝑃 are the solution to (1.1).
(b) If 𝐴 is an idempotent operator satisfying 𝐴𝐵=𝐵𝐴, then 𝐵 is one solution to (1.1) if and only if there exists a square-zero operator 𝑁0 such that 𝐵=𝐴+𝑁0 and 𝐴𝑁0=𝑁0𝐴=0.
(c) If 𝐴 is an orthogonal projection, then self-adjoint operator 𝐵 satisfies (1.1) if and only if 𝐴=𝐵.

Proof. (a) See Theorem 2.2 in [3].
(b) By simultaneous similarity transformations, 𝐴 and 𝐵 can be written as 𝐴=𝐼⊕0 and 𝐵=𝑁21⊕𝑁22 since 𝐴𝐵=𝐵𝐴. From 𝐴𝐵𝐴=𝐴2, we obtain 𝑁21=I. From 𝐵𝐴𝐵=𝐵2, we obtain 𝑁222=0. Select 0⊕𝑁22=𝑁0. Then 𝑁20=0, 𝐵=𝐴+𝑁0 and 𝐴𝑁0=𝑁0𝐴=0.
(c) We use the notations from Theorem 3.3. If 𝐴 is an orthogonal projection, then 𝐴1=𝐼,𝐵2=0 in the proof of Theorem 3.3, so the result is a direct corollary of Theorem 3.3.

4. The Perturbation of the Solutions

The operators 𝐴 and 𝐵 are said to be c-orthogonal, denoted by 𝐴⟂𝑐𝐵, whenever 𝐴𝐵=0 and 𝐵𝐴=0. The next result is a generalization of Theorems 2.2 and 3.2 in [6], where the same problems have been considered for ind(𝐴)≤1 and ind(𝐵)≤1.

Theorem 4.1. Let 𝐴 be generalized Drazin invertible such that 𝐴𝜋𝐵(𝐼−𝐴𝜋)=0. Then 𝐴and𝐵satisfy(1.1)iff𝐴=𝑃1+𝑁1,𝐵=𝑃2+𝑁2,(4.1) where 𝑁1 and 𝑁2 are arbitrary quasinilpotent elements satisfying (1.1), 𝑃1 and 𝑃2 are arbitrary idempotent elements satisfying ℛ(𝑃1)=ℛ(𝑃2) and 𝑃𝑖⟂𝑐𝑁𝑗, 𝑖,𝑗=1,2.

Proof. Let us consider the matrix representation of 𝐴 and 𝐵 relative to the 𝑃=𝐼−𝐴𝜋. We have 𝐴𝐴=100𝐴2𝑃𝐵,𝐵=1𝐵2𝐵3𝐵4𝑃,(4.2) where 𝐴1 is invertible and 𝐴2 is quasinilpotent. From 𝐴𝐵𝐴=𝐴2, we have 𝐵1=𝐼,𝐵2𝐴2=0,𝐴2𝐵3=0,𝐴2𝐵4𝐴2=𝐴22.(4.3) From 𝐵𝐴𝐵=𝐵2 and (4.3), we have 𝐼+𝐵2𝐵3=𝐴1,𝐵2+𝐵2𝐵4=𝐴1𝐵2,𝐵3+𝐵4𝐵3=𝐵3𝐴1,𝐵3𝐵2+𝐵24=𝐵3𝐴1𝐵2+𝐵4𝐴2𝐵4.(4.4) From 𝐴𝜋𝐵(𝐼−𝐴𝜋)=0, we have 𝐵3=0. Now, it follows from (4.2), (4.3) and (4.4) that 𝐴=𝐼00𝐴2𝑃,𝐵=𝐼𝐵20𝐵4𝑃,(4.5) with 𝐴2𝐵4𝐴2=𝐴22,𝐵4𝐴2𝐵4=𝐵24,𝐵2𝐴2=0,𝐵2𝐵4=0.(4.6) Hence, 𝐵4 is quasinilpotent by Lemma 2.6. Select 𝑁1=0⊕𝑃𝐴2,𝑁2=0⊕𝑃𝐵4,𝑃1=𝐼⊕𝑃0,𝑃2=𝐵−𝑁2.(4.7) Then 𝑃1 and 𝑃2 are idempotent operators and ℛ(𝑃1)=ℛ(𝑃2). 𝑁1 and 𝑁2 are quasinilpotent operators satisfying (1.1) and 𝑃𝑖⟂𝑐𝑁𝑗,𝑖,𝑗=1,2.
For the proof of sufficiency observe that ℛ(𝑃1)=ℛ(𝑃2) leads to 𝑃1𝑃2=𝑃2,𝑃2𝑃1=𝑃1. Straightforward calculations show that 𝐴𝐵𝐴=𝐴2 and 𝐵𝐴𝐵=𝐵2.

We also prove the next result which can be seen as one corollary of Theorem 4.1.

Corollary 4.2. Let 𝐴 be generalized Drazin invertible such that 𝐴𝜋𝐴𝐵=𝐴𝜋𝐵𝐴. Then 𝐴and𝐵satisfy(1.1)iff𝐴=𝑃1+𝑁1,𝐵=𝑃2+𝑁2,(4.8) where 𝑁1 and 𝑁2 are 2-nilpotent operators satisfying 𝑁1𝑁2=𝑁2𝑁1, 𝑃1 and 𝑃2 are arbitrary idempotent elements satisfying ℛ(𝑃1)=ℛ(𝑃2) and 𝑃𝑖⟂𝑐𝑁𝑗,𝑖,𝑗=1,2.

Proof. The sufficiency is clear. For the proof of the necessity, let 𝐴 and 𝐵 have the matrix representation as (4.2). From 𝐴𝐵𝐴=𝐴2 and 𝐵𝐴𝐵=𝐵2, we know that 𝐴𝑖,𝑖=1,2 and 𝐵𝑖,𝑖=1,2,3,4 satisfy (4.3) and (4.4). The condition 𝐴𝜋𝐴𝐵=𝐴𝜋𝐵𝐴 implies that 𝐴2𝐵3=𝐵3𝐴1,𝐴2𝐵4=𝐵4𝐴2.(4.9) It follows that 𝐵3=0 because 𝐴2𝐵3=0 and 𝐴1 is invertible by (4.3) and (4.4). Also, (4.2), (4.3), and (4.4) imply 𝐴2 is quasinilpotent and 𝐴22=𝐴2𝐵4𝐴2,𝐵24=𝐵4𝐴2𝐵4,𝐵4𝐴2=𝐴2𝐵4.(4.10) It follows immediately that 𝐴22=𝐵24=0 by Lemma 2.6. Now, we obtain 𝐴=𝐼00𝐴2𝑃,𝐵=𝐼𝐵20𝐵4𝑃,(4.11) with 𝐵2𝐵4=0,𝐵2𝐴2=0,𝐴2𝐵4=𝐵4𝐴2,𝐴22=𝐵24=0.(4.12) Select 𝑁1=0⊕𝑃𝐴2,𝑁2=0⊕𝑃𝐵4,𝑃1=𝐼⊕𝑃0,𝑃2=𝐵−𝑁2.(4.13) Then 𝑃1 and 𝑃2 are idempotent operators and ℛ(𝑃1)=ℛ(𝑃2). 𝑁21=0 and 𝑁22=0 satisfying 𝑃𝑖⟂𝑐𝑁𝑗, 𝑖,𝑗=1,2 and 𝑁1𝑁2=𝑁2𝑁1.

If we assume that 𝐴𝐵=𝐵𝐴 instead of the condition 𝐴𝜋𝐴𝐵=𝐴𝜋𝐵𝐴, we will get a much simpler expression for 𝐴 and 𝐵.

Corollary 4.3. Let 𝐴 be generalized Drazin invertible such that 𝐴𝐵=𝐵𝐴. Then 𝐴and𝐵satisfy(1.1)iff𝐴=𝑃+𝑁1,𝐵=𝑃+𝑁2,(4.14) where 𝑁1 and 𝑁2 are 2-nilpotent operators satisfying 𝑁1𝑁2=𝑁2𝑁1, 𝑃 is arbitrary idempotent element satisfying 𝑃⟂𝑐𝑁1 and 𝑃⟂𝑐𝑁2.

Proof. Similar to the proof of Theorem 4.1, Corollary 4.2. If 𝐴𝐵=𝐵𝐴, then 𝐴 and 𝐵 have the matrix representations 𝐴=𝐼⊕𝑃𝐴2,𝐵=𝐼⊕𝑃𝐵4,(4.15) with 𝐴22=𝐵24=0 and 𝐵4𝐴2=𝐴2𝐵4, so, by Corollary 4.2, the result is clear.

Remark 4.4. (1) Let 𝛾(𝑇) denote the spectrum radius of operator 𝑇. In Corollaries 4.2 and 4.3, since nilpotent operators 𝐴2 and 𝐵4 are commutative, we get 𝛾𝐴2−𝐵4𝐴≤𝛾2+𝛾−𝐵4𝐵=0,𝛾4𝐴2𝐵≤𝛾4𝛾𝐴2=0,(4.16) that is, 𝐴2−𝐵4 and 𝐵4𝐴2 are nilpotent. Hence, 𝐴−𝐵 is a nilpotent operator and 𝐵𝐴 can be decomposed as the 𝑐-orthogonality sum of an idempotent operator and a 2-nilpotent operator.
(2) Let 𝐴 have the Drazin inverse 𝐴𝑑. Then, by (4.2), 𝐴 can be written as 𝐴=𝐴1⊕𝐴2, where 𝐴1 is invertible and 𝐴2 is quasi-nilpotent (see also [4, 5]). If 𝐴=𝐴∗, then𝐴𝑑=𝐴#=𝐴+=𝐴1−1⊕0,(4.17) where 𝐴+ is the Moore-Penrose inverse of 𝐴 and 𝐴# is the group inverse. In fact, if 𝐴=𝐴∗, then 𝐴2=0 because the self-adjoint quasi-nilpotent operator must be zero. Hence 𝐴=𝐴1⊕0,𝐴𝑑=𝐴#=𝐴+=𝐴1−1⊕0.
(3) If self-adjoint operators 𝐴 and 𝐵 satisfy (1.1), then ‖𝐴‖≥1,𝐴𝐵=𝐵𝐴iff‖𝐴‖=1iff𝐴=𝐵=𝑃𝒩(𝐴)⟂,(4.18) where 𝑃𝒩(𝐴)⟂ is the orthogonal projection on 𝒩(𝐴)⟂. In fact, let ℋ=𝒩(𝐴)⟂⊕𝒩(𝐴), then 𝐴=𝐴1⊕0. Select 𝐵𝐵=1𝐵2𝐵∗2𝐵4.(4.19) Similar to the proof of Theorem 4.1, we have 𝐵1=𝐼,𝐴1=𝐼+𝐵2𝐵∗2,𝐵∗2𝐴1𝐵2=𝐵∗2𝐴1𝐵2+𝐵24.(4.20) This shows that ‖𝐴‖≥1 since 𝐴1=𝐼+𝐵2𝐵∗2≥𝐼. If ‖𝐴‖=1, then 𝐴1=𝐼,𝐵2=0,𝐵4=0. Hence 𝐴=𝐵=𝑃𝒩(𝐴)⟂. If 𝐴𝐵=𝐵𝐴, then 𝐵2=0, so 𝐴=𝐼⊕0,𝐵=𝐼⊕𝐵4. From 𝐵𝐴𝐵=𝐵2, we have 𝐵24=0, so 𝐵4=0 since 𝐵4=𝐵∗4. Hence 𝐴=𝐵=𝑃𝒩(𝐴)⟂. These results (see Theorem 4.2 in [6]) can be seen as the particular case of Corollary 4.3.

Acknowledgments

The author would like to thank the anonymous referees for their careful reading, very detailed comments, and many constructive suggestions which greatly improved the paper. C. Y. Deng is supported by a Grant from the Ph.D. Programs Foundation of Ministry of Education of China (no. 20094407120001).

References

  1. I. Vidav, “On idempotent operators in a Hilbert space,” Publications de l'Institut Mathématique. Nouvelle Série, vol. 4, no. 18, pp. 157–163, 1964. View at: Google Scholar | Zentralblatt MATH
  2. V. Rakočević, “A note on a theorem of I. Vidav,” Publications de l'Institut Mathématique. Nouvelle Série, vol. 68, no. 82, pp. 105–107, 2000. View at: Google Scholar | Zentralblatt MATH
  3. C. Schmoeger, “On the operator equations ABA=A2 and BAB=B2,” Publications de l'Institut Mathématique. Nouvelle Série, vol. 78, no. 92, pp. 127–133, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  4. A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, Wiley-Interscience, New York, NY, USA, 1974. View at: Zentralblatt MATH
  5. A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, Springer, New York, NY, USA, 2nd edition, 2002.
  6. C. Y. Deng, “The Drazin inverses of sum and difference of idempotents,” Linear Algebra and Its Applications, vol. 430, no. 4, pp. 1282–1291, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. J. J. Koliha and V. Rakocevic, “Range projections and the Moore-Penrose inverse in rings with involution,” Linear and Multilinear Algebra, vol. 55, no. 2, pp. 103–112, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  8. G. J. Murphy, C*-Algebra and Operator Theory, Academic Press, New York, NY, USA, 1990.
  9. H. K. Du, “Operator matrix forms of positive operator matrices,” Chinese Quarterly Journal of Mathematics, vol. 7, pp. 9–11, 1992. View at: Google Scholar
  10. D. T. Kato, Perturbation Theory for Linear Operators, Springer, New York, NY, USA, 1966.
  11. P. R. Halmos, “Two subspaces,” Transactions of the American Mathematical Society, vol. 144, pp. 381–389, 1969. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  12. C. Y. Deng and H. Du, “A new characterization of the closedness of the sum of two subspaces,” Acta Mathematica Scientia, vol. 28, no. 1, pp. 17–23, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH

Copyright © 2011 Chun Yuan Deng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views876
Downloads362
Citations

Related articles

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.