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ISRN Applied Mathematics

Volume 2012 (2012), Article ID 251389, 13 pages

http://dx.doi.org/10.5402/2012/251389

## Maps Completely Preserving Involutions and Maps Completely Preserving Drazin Inverse

^{1}College of Science, Harbin Engineering University, Harbin 150001, China^{2}Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China^{3}East University of Heilongjiang, Harbin 150001, China

Received 24 August 2012; Accepted 2 October 2012

Academic Editors: H. Y. Chung and H. A. Larrondo

Copyright © 2012 Hongmei Yao et al. 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.

#### Abstract

Let and be infinite dimensional Banach spaces over the real or complex field , and let and be standard operator algebras on and , respectively. In this paper, the structures of surjective maps from onto that completely preserve involutions in both directions and that completely preserve Drazin inverse in both direction are determined, respectively. From the structures of these maps, it is shown that involutions and Drazin inverse are invariants of isomorphism in complete preserver problems.

#### 1. Introduction

In the last decades, the study of preserver problems is an active topic in operator algebra or operator space theory (see [1]). In [2], the form of involutivity-preserving maps was given by using the known results of idempotence-preserving maps, and in [3], the authors gave the characterization of additive maps preserving Drazin inverse. These results showed that involutions and Drazin inverse are invariants of isomorphism in preserver problems. Since completely positive linear maps and completely bounded linear maps are very important in operator algebra or operator space theory [4], and the concept of completely rank nonincreasing linear maps was introduced by Hadwin and Larson in [5], many mathematicians began to focus on complete preserver problems, that is, characterizations of maps on operator spaces (subsets) that preserve some property (or invariant) completely [6]. Cui and Hou discussed the completely trace-rank-preserving linear maps and the completely invertibility-preserving linear maps in [7, 8], respectively. Subsequently, in [6, 9], general surjective maps between standard operator algebras that completely preserve invertibility or spectrum and that completely preserve spectral functions are studied, respectively, where a standard operator algebra is a norm closed subalgebra of some over a Banach space containing the identity and all finite-rank operators. Recently, in [10], the authors discussed completely idempotents preserving surjective maps and completely square-zero operators preserving surjective maps. These results showed that idempotents and square-zero operators are invariants of isomorphism in complete preserver problems. Since involutions and Drazin inverse are closely related to idempotents, it is interesting to consider whether the involutions and Drazin inverse are still invariants of isomorphism in complete preserver problems.

Let and be Banach spaces over the real or complex field , and let be the Banach algebra of all bounded linear operators from to . An operator is called an involution (idempotent) if (), denoted by and , where is an algebra and is an identity in . An operator is said to have a Drazin inverse, or to be Drazin invertible if there exists such that and is called the Drazin inverse of , denoted by . The concepts of involution and Drazin inverse are very useful in various applied mathematical areas. For example, in [11], the authors showed that involution has applications in Chi-square distribution, combinatorial problems, and so on. About Drazin inverse, it is helpful in singular differential and difference equations, Markov chain, multibody system dynamics, and so on [3].

Inspired by the above, the purpose of this paper is to consider the following two things: (1)the characterization of surjective maps that completely preserve involutions between standard operator algebras on Banach spaces; (2)the characterization of surjective maps that completely preserve Drazin inverse between standard operator algebras on Banach spaces.

Let and be standard operator algebras on and , respectively, and let be a surjective map. Define, for each , a map by Then is called -involutions preserving in both directions if preserves involutions in both directions, that is, ; is said to be completely involution-preserving in both directions if is -involutions preserving in both directions for every positive integer . Similarly, is called -Drazin inverse preserving in both directions if preserves Drazin inverse in both directions, that is, ; is said to be completely Drazin inverse preserving in both directions if is -Drazin inverse preserving in both directions for every positive integer .

We end this part by some notations. Let be the dual space of a Banach space . For every nonzero and , the symbol standards for the rank one bounded linear operator on defined by for any . Given , we say and are orthogonal if , where is the zero operator in .

#### 2. Maps Completely Preserving Involutions

Lemma 2.1 (see [10]). *Let be infinite dimensional Banach spaces over the real or complex field and , be sets of idempotents which contain all rank one idempotents. Let be a bijective map. If preserves orthogonality in both directions, then there exists either a bounded invertible linear or (in the complex case) conjugate linear operator such that
**
or a bounded invertible linear or (in the complex case) conjugate linear operator such that
**
In the second case, and must be reflexive. *

Theorem 2.2. *Let be infinite-dimensional Banach spaces over the real or complex field and be standard operator algebras on and , respectively. Let be a surjective map. Then the following statements are equivalent: *(1)* is completely involutions preserving in both directions. *(2)* is -involution preserving in both directions. *(3)*There exists a bounded invertible linear or (in the complex case) conjugate linear operator such that
**where . *

* Proof. *Obviously, . Then, is shown by proving the following claims. Assume that is -involution preserving in both directions. *Claim 1. *, , where and is injective.

For any ,
where is a Banach space with a suitable norm, for example, . Applying the assumption of , we get
Thus
By the surjectivity of , we can find some such that . Let , (2.6) yields that . Hence, holds for all ; this entails that , since is surjective.

Taking in (2.7), and also by the invertibility of , we have . Then (2.7) yields that for all . Because of the surjectivity of and , it is not difficult to get , where .

If we replace by , it is still -involution preserving, then without loss of generality, we always assume that in the sequel. Next, we show that is injective.

For any such that , we have
which imply that . Therefore, is injective.*Claim 2. * preserves idempotents in both directions.

For any , since
then using the assumption of , we have
From (2.10) and (2.11), it is derived that for any . Applying (2.11) again, we see that
Since is arbitrary, then (2.12) yields that
Thus, combining (2.13) and (2.14) with the bijectivity of , it is not difficult to get the result that if , then . Therefore, Claim 2 holds true.*Claim 3.* There exists a bounded invertible linear or (in the complex case) conjugate linear operator such that for every .

For every ,
that is, preserves orthogonality in both directions from to . From Lemma 2.1, we see that there exists either a bounded invertible linear or (in the complex case) conjugate linear operator such that
or a bounded invertible linear or (in the complex case) conjugate linear operator such that

Sequently, we show that the second case cannot occur. On the contrary, assume that for all . Similar to the proof of Theorem 3.2 in [10], for any linearly independent vectors , there exist such that and . Then,
By the assumption of on and (2.14), we see that
but is not an involution, it is a contradiction. Therefore, Claim 3 holds true.

Let , then is a bijective map preserving 2 involutions in both directions from onto the standard operator algebra . Furthermore, by Claim 3, for every . Hence, without lose of generality, we suppose that
for all .*Claim 4. * for any rank one operator .

For any rank one operator , there exists such that is linearly independent of and . Then there exist such that , . Let , then we have . Then using (2.20), we have
Hence,
Combining (2.22) and (2.23) with (2.24), we derive that . Then using (2.22) and (2.23) again, we get
From (2.25), it is easily seen that there exists such that
Taking (2.26) into (2.22), this yields that , Thus, Claim 4 holds true.*Claim 5. * for all .

For any , since
By , it follows that

For any and any invertible operator ,
Applying (2.28), we get
For any rank one operator , let in (2.30), and using Claim 4, we know that . It follows that . This yields that

For any rank one operator , it is clearly that is either idempotent or invertible. Then using (2.20) or (2.31), we have

For any ,
By (2.28), we have
For any rank one operator , let in (2.34), and using (2.32), we still get . Then similarly, we have
Therefore, the proof of this theorem is finished.

is called -identity product preserving in both directions if preserves identity product in both directions, that is, ; is said to be completely identity preserving product in both directions if is -identity product preserving in both directions for every positive integer and is called -identity Jordan product preserving in both directions if preserves identity Jordan product in both directions, that is, ; is said to be completely identity Jordan product preserving in both directions if is -identity Jordan product in both directions for every positive integer .

*Remark 2.3. *Using the result of Theorem 2.2, it is not difficult to give the characterization of maps completely preserving identity product in both directions and maps completely preserving identity Jordan product preserving in both directions.

#### 3. Maps Completely Preserving Drazin Inverse

Theorem 3.1. *Let be infinite-dimensional Banach spaces over the real or complex field and be standard operator algebras on and , respectively. Let be a surjective map. Then the following statements are equivalent: *(1)* is completely Drazin inverse preserving in both directions. *(2)* is -Drazin inverse preserving in both directions. *(3)*There exists a bounded invertible linear or (in the complex case) conjugate linear operator such that
where . *

* Proof. *Clearly, we only need to prove that . Assume that is -Drazin inverse preserving in both directions.*Claim 1. *, , and is injective.

For any , since
using (1.1), it entails that
As is surjective, there exists some such that . Taking in (3.4) and (3.5), respectively, we have
Taking (3.6) and (3.7) into (3.4) again, we see that
then let in (3.8) and use (3.7), we get
Taking (3.9) into (3.3), this yields that
by the surjectivity of , there exists a such that ; let in (3.10), we see that
Since
then by the assumption of and applying (1.1) and (3.11), we see that
Let in (3.13), we have
For any ,
then using (3.14) and (3.11), we have
Similar to the proof of Claim 1 in Theorem 2.2, we get , where .

Without loss of generality, we always assume that in the sequel. Now, we show that is injective.

Take into (3.13), it yields that
For any such that , we have
then
therefore, by (3.17) and , we see that
then
Applying (1.1), we derive that , and . By direct computation, it is easy to get . Thus, is an injective map, and Claim 1 holds true.*Claim 2. * preserves idempotents in both directions.

For any ,
by the assumption of and (1.1), we have

For any ,
it follows that and . Then by (3.23), it derives that . Similarly, we get that if , then . Therefore, this claim is true.*Claim* 3*.* There exists a bounded invertible linear or (in the complex case) conjugate linear operator such that for every .

For every , ,
that is, preserves orthogonality in both directions from to . It follows from Lemma 2.1 that there exists either a bounded invertible linear or (in the complex case) conjugate linear operator such that
or a bounded invertible linear or (in the complex case) conjugate linear operator such that

We show that the second case cannot occur. On the contrary, assume that for all . For any linearly independent vectors , similar to the proof of the Theorem 3.2 in [10], we can find such that and . Then
but by the assumption of and (1.1), it is easy to check that
which is a contradiction to the hypothesis that is Drazin inverse preserving. Therefore, holds for every , and Claim 3 holds true.

In the sequel, without lose of generality, we suppose that
*Claim *4. for any rank one operator .

Similar to the proof of Claim 4 in Theorem 2.1 in [10], for any rank one operator , we can find such that and such that . Then
Using (3.30), we have
Therefore, we derive that and . Then, it is easily seen that there exists such that
Similar to the proof of Claim 4 in Theorem 2.2, for any rank one operator , we can find such that and such that . Then
by (3.30) and the assumption of , we see that
It entails that
by (3.33), we know that . Therefore, Claim 4 holds true.*Claim* 5*. * for all .

For any , since
then by the assumption of , using (3.17) and (3.23), we have
For simplification, let ; then by (1.1), we derive that and . Therefore, we have . Thus, by direct computation, we get
For any rank one operator , let in (3.39), and using Claim 4, we have . It follows that . Then we get
Therefore, the proof of this theorem is completed.

#### Acknowledgments

The authors show great thanks to the referee for his/her valuable comments, which greatly improved the readability of the paper. This work was mainly supported by the National Natural Science Foundation Grants of China (Grant no. 10871056) and the Fundamental Research Funds for the Central Universities (Grant no. HEUCF20111132).

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