Maps Preserving Peripheral Spectrum of Generalized Jordan Products of Self-Adjoint Operators

Zhang, Wen; Hou, Jinchuan

doi:https://doi.org/10.1155/2014/192040

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2014 | Article ID 192040 | https://doi.org/10.1155/2014/192040

Maps Preserving Peripheral Spectrum of Generalized Jordan Products of Self-Adjoint Operators

Wen Zhang¹and Jinchuan Hou^1,2

Academic Editor: Márcia Federson

Received09 Feb 2014

Accepted25 Aug 2014

Published19 Oct 2014

Abstract

Let and be standard real Jordan algebras of self-adjoint operators on complex Hilbert spaces and , respectively. For , let be a fixed sequence with and assume that at least one of the terms in appears exactly once. Define the generalized Jordan product on elements in . Let be a map with the range containing all rank-one projections and trace zero-rank two self-adjoint operators. We show that satisfies that for all , where stands for the peripheral spectrum of , if and only if there exist a scalar and a unitary operator such that for all , or for all , where is the transpose of for an arbitrarily fixed orthonormal basis of . Moreover, whenever is odd.

1. Introduction

Recently, the study of maps preserving spectrum of products of operators attracted attentions of researchers. In [1], Molnár characterized surjective maps on bounded linear operators acting on a Hilbert space preserving the spectrum of the product of operators; that is, and always have the same spectrum. This similar question was studied by Huang and Hou in [2] by replacing the spectrum by several spectral functions such as the left spectrum and spectral boundary. Hou et al. [3, 4] studied, respectively, further the maps between certain operator algebras preserving the spectrum of a generalized product and a generalized Jordan product of low rank operators. Note that the linear maps between Banach algebras which preserve the spectrum are extensively studied in connection with a longstanding open problem due to Kaplansky on invertibility preserving linear maps ([5–10] and the references therein).

Moreover, there has been considerable interest in studying peripheral spectrum preserving maps on operator algebras. Recall that the peripheral spectrum of an element in a complex Banach algebra is defined by where and stand for the spectrum and the spectral radius of , respectively. Recall also that a set-valued map is said to be a spectral function if for every . Since is compact, is a well-defined nonempty set and is an important spectral function. Observe that it is always true that . In [11], Tonev and Luttman studied maps preserving peripheral spectrum of the usual operator products on standard operator algebras. Recall that a standard operator algebra is a subalgebra of that contains the identity and all finite rank operators, where stands for as usual the Banach algebra of all bounded linear operators on Banach space . They studied also the corresponding problems in uniform algebras (see [12, 13]). Miura and Honma [14] generalized the result in [13] and characterized surjective maps and satisfying on standard operator algebras. Cui and Li studied in [15] the maps preserving peripheral spectrum of Jordan products of operators on standard operator algebras. In [16] the maps preserving peripheral spectrum of Jordan semitriple products of operators were characterized. The authors studied in [17, 18], respectively, further the maps between certain operator algebras which preserve peripheral spectrum of a generalized product and a generalized Jordan product as defined below.

Definition 1. Fix a positive integer and a finite sequence such that and there is an not equal to for all other ; that is, appears just one time in the sequence. For operators , the operators, are, respectively, called generalized product and generalized Jordan product of , while is called the width of the products.

Evidently, the generalized Jordan product (the generalized product ) covers the Jordan product and the Jordan triple product (the usual product and the Jordan semitriple product ), and so forth. We also remark that the notations and are not unique for because they depend on the choice of the integers , , and the sequence . In this paper, we presume that , , and the sequence are arbitrary but fixed throughout the paper.

Let us consider the case of Hilbert spaces. Denote by the set of all bounded linear operators on a complex Hilbert space and the adjoint of . If , is self-adjoint. Denote by the real Jordan algebra of all self-adjoint operators in . A real Jordan subalgebra of is said to be standard if it contains the identity and all finite rank self-adjoint operators. In [14] Miura and Honma characterized the surjective maps between standard operator algebras on Hilbert spaces that preserve the peripheral spectrum of skew products of operators. Cui and Li studied in [15] the maps preserving peripheral spectrum of skew Jordan products of operators on standard operator algebras on complex Hilbert spaces. A characterization of maps preserving peripheral spectrum of Jordan products of self-adjoint operators on standard real Jordan subalgebras of was also given in [15]. In [16] the maps preserving peripheral spectrum of Jordan skew semitriple products of operators were characterized, and then, the maps preserving peripheral spectrum of the skew generalized products of operators on Hilbert space were characterized in [17].

Products of self-adjoint operators in Hilbert space play a role in several different areas of pure and applied mathematics. In this paper, we characterize the maps preserving the peripheral spectrum of generalized Jordan products of self-adjoint operators between the standard real Jordan algebras of self-adjoint operators on complex Hilbert spaces. Let be a standard real Jordan algebra in , , and a map with range containing all rank-one projections and all rank-two self-adjoint operators with zero trace. We show that satisfies that for all in if and only if there exist a scalar and a unitary operator such that for all , or for all , where is the transpose of with respect to an arbitrary but fixed orthonormal basis of . Moreover, whenever is odd. We also characterize the maps from into that preserves the peripheral spectrum of generalized product on .

2. Generalized Jordan Products of Self-Adjoint Operators

Let and be two complex Hilbert spaces and and the real linear spaces of all self-adjoint operators in and , respectively. Then and are real Jordan algebras. Recall that a standard real Jordan algebra on is a Jordan subalgebra of which contains all finite rank self-adjoint operators and the identity operator. In this section, we will characterize maps preserving peripheral spectrum of generalized Jordan products of self-adjoint operators.

Theorem 2. Let and be standard real Jordan algebras of self-adjoint operators on complex Hilbert spaces and , respectively. Consider the product defined in (3) of Definition 1 with the width . Assume that is a map the range of which contains all rank-one projections and all rank-two self-adjoint operators with zero trace. Then satisfies for all if and only if there exist a unitary operator and a scalar such that either(1) for every , or(2) for every . Here is the transpose of with respect to an arbitrary but fixed orthonormal basis of .
Moreover, whenever is odd.

To prove Theorem 2, we consider the special case that and for all . Thus there exist nonnegative integers , with such that . For this special case we have.

Theorem 3. Let and be standard real Jordan algebras of self-adjoint operators on complex Hilbert spaces and , respectively. Assume that is a map the range of which contains all rank-one projections and all rank-two self-adjoint operators with zero trace, and are nonnegative integers with . Then satisfies for all if and only if there exist a unitary operator and a scalar such that for every , or for every . Moreover, whenever is even. Here is the transpose of with respect to an arbitrary but fixed orthonormal basis of .

If meets (4), then it also meets (5) for some , with by taking and for . Hence it is obvious that the truth of Theorem 3 will imply the truth of Theorem 2.

Thus we focus our attention to prove Theorem 3. We will do it by decomposing the proof in a number of steps and use of technical lemmas.

Note that, if , then the question is reduced to the generalized product of self-adjoint operators, which will be discussed in the next section. So, unless specified otherwise, we always assume in this section that .

Lemma 4. For any unit vector and nonzero , we have

Proof. In fact, if there exist nonzero such that , , clearly (6) holds. Now assume that and or and are linearly independent. Then there exist nonzero and such that ; that is, It follows that We consider the following two cases.
Case 1 . If , it follows from (10) that . Then (9) and (10) imply that . If , it follows from (9) that , but this contradicts (7). So .
Case 2 . In this case, there must be and . Then it follows from (9) and (10) that which implies that . So Now the result follows immediately.

In Lemmas 5 and 6, we always assume that is a map satisfying (5) with range containing all rank-one projections and all rank-two self-adjoint operators of zero trace, and assume that , are nonnegative integers with . Recall that a self-adjoint operator is said to be positive, denote by , if for all ; while means that .

Lemma 5. or . may occur only if is odd.

Proof. For any , since it follows that holds for every . Let . By the assumption on the range of , for any unit vector , there exists such that . We consider the following two cases.
Case 1 . It follows from (5) that which implies that for all unit vectors . Then by Lemma 4, we have and hence for all unit vectors , and . On the other hand, for any unit vector , we have . Hence holds for all unit vectors and consequently, . So we must have and for all unit vectors . Now it follows from (15) that with . In particular, if is even, as , (14) and (15) imply that for all unit vectors and hence .
Case 2 . By (5) we have which implies that for all unit vectors . Then holds for each unit vector , and so with . Particularly, if is even, then and it follows from (16) that holds for every unit vector . Hence .

If , then satisfies the conditions in Theorem 3, so we may as well assume in the sequel, and thus holds for every .

Lemma 6. preserves rank-one projections in both directions.

Proof. We consider the following two cases.
Case 1 . Consider the following.
Case 1.1 ( is even). For any unit vector , let and . It follows from that .
Note that if , and , then . If fact implies that . Since , we must have and , which forces .
Now, as and , we see that . It follows from that , which implies that , where ran stands for the range of . For any unit vector , pick such that . It follows from that , which, together with , implies that . So we have and is rank-one.
Case 1.2 ( is odd). For any unit vector , let and . We will prove that is a rank-one projection.
Claim 1.2.1 . Note that . Then . It follows that either (i) or (ii) is injective but not surjective.
Assume that (ii) occurs. Since , we have and . So, according to some suitable space decomposition of , has an operator matrix representation of the form where and . To see this, one can first choose three orthonormal vectors , , such that for some sufficiently small . Suppose the compression of onto the span of has eigenvalues . Then Let . Then is similar to Thus, there exists a space decomposition such that has an operator matrix of the form Clearly, there are unitary , such that has operator matrix of the form where . So has the desired operator matrix form. Under the same decomposition, take ; then has two different points with and there exists such that . It follows that . So and for all unit vectors . But is either a singleton or with . This contradicts the fact .
So . Assume that . According to the space decomposition , has an operator matrix . Under the same space decomposition, take . Similar to the previous discussion, one gets a contradiction again. So .
Claim 1.2.2. There exists a unit vector such that .
If it is not true, then, by Claim , there exist a unit vector and a nonzero with such that . So there exists a unit vector such that . Let and . Then , , and . Since the range of contains all rank-one projections, there exist and in such that and . Then , , , and .
Since , it follows from (6) that , which, together with , implies that . So, according to the space decomposition , with . If has an operator matrix accordingly, then Since is invertible, we see that . Clearly, . So, . But then it contradicts the fact that . So Claim holds and preserves rank-one projections.
Conversely, assume that is a rank-one projection; then a similar discussion shows that is a rank-one projection, too.
Case 2 .Consider the following.
Case 2.1 ( is even). For any unit vector , let and . It follows from that . Since and , we see that . If , then and , which, together with , imply that , and thus , a contradiction.
So, . Take a unit vector and such that . It follows from that , which, together with , implies that . Hence we have and .
Case 2.2 ( is odd). For any unit vectors , let and . We will prove that is a rank-one projection.
Claim 2.2.1 . Note that . Then . It follows that either (i) or (ii) is injective but not surjective.
Assume that (ii) occurs. Since , we have and . So, like shown in Case , with respect to some suitable space decomposition of , has an operator matrix representation of the form where and . Under the same decomposition, take , and then . As has rank-two and zero trace, there exists such that . It follows that . So and for all unit vectors . But . This contradicts the fact .
So . Assume that . According to the space decomposition , has an operator matrix . Under the same space decomposition, take . Similar to the previous discussion, one gets a contradiction again. So .
Claim 2.2.2. There exists a unit vector such that .
If it is not true, then, by Claim , there exist a unit vector and a nonzero with such that . So there exists a unit vector such that and . Let and . Since the range of contains all rank-one projections, there exist and in such that and . Then , which, together with (6), implies that . It follows from that . So , and according to the space decomposition , with . Thus under the same space decomposition we have with . Write in the operator matrix accordingly; then Clearly, . So, . But then this contradicts the fact that . So Claim holds and preserves rank-one projections.
Conversely, assume that is a rank-one orthogonal projection; then, a similar discussion implies that is a rank-one projection.

The following lemma was proved in [19].

Lemma 7. Let be a complex Hilbert space and self-adjoint operators. If holds for all , then .

Now we are in a position to present our proof of Theorem 3, except the case .

Proof of Theorem 3. The “if” part is obvious. Let us check the “only if” part.
By Lemma 6, preserves rank-one projections in both directions. It follows that there exists a bijective map such that for all unit vectors , where and for any and .
We consider the following two cases.
Case 1 . For any unit vectors , we have . By (6), if and only if , and when , It follows that holds for all .
Wigner’s theorem [20] states that every bijective map between Hilbert spaces , satisfying (27) must have the form for any , where is a unitary or a conjugate unitary (i.e., antiunitary) operator and is a so-called phase-function which means that its values are of modulus one. Thus, by Wigner’s theorem, there exists a unitary or conjugate unitary operator such that for every unit vector .
Assume first that is unitary. Let be arbitrary. For any unit vector , Applying (6), for any unit vector , one has and hence Lemma 7 implies that . Hence, for every . We claim that for every . Otherwise, there exists some nonzero such that . Let and . Then , , and . Note that, as , . It follows that, for any , we have . Therefore, holds for all . Let be the spectral resolution of . Then the spectral project . Though we do not know if , we can take unit vector so that . Thus, . By Lemma 4, we have . Since and , . But then, a contradiction. So, holds for every .
Now assume that is conjugate unitary. Take arbitrarily an orthonormal basis of and define by . Then is conjugate unitary and . Let . Then is unitary and a similar discussion as above implies that for all and is the transpose of for the orthonormal basis of .
Case 2 . For any unit vectors , we have Hence holds for all . Thus, by Wigner’s theorem, there exists a unitary or conjugate unitary operator such that for every unit vector .
Now assume that is unitary. Let be arbitrary. For any unit vector , since we have So we get for every .
Similar to the case , if is conjugate unitary, then there exists a unitary operator such that for all .
Hence we have shown that, in the case , there exists a unitary such that either for every ; or for every , where is the transpose of with respect to an arbitrarily given orthonormal basis of .
If , considering gives for every or for every . It is clear that this case does not occur if is even.

3. Generalized Products of Self-Adjoint Operators on Hilbert Spaces

In this section, we will characterize maps preserving peripheral spectrum of generalized products of self-adjoint operators. Its special case, Theorem 10, makes up for the gap for the case in the proof of Theorem 3.

Let be a real Jordan algebra in . If a generalized product defined in (2) satisfies that for any , that is, the general product is closed in , we say that is a generalized product on . The following lemma was proved in [3].

Lemma 8. Let be a generalized product on a standard real Jordan algebra defined as in (2) of Definition 1. Then there exists a positive integer with such that , and for all .

The following is the main result in this section. Observe that we do not need the assumption that the range of the map contains all rank-two self-adjoint operators with zero trace.

Theorem 9. Let and be standard real Jordan algebras of self-adjoint operators on complex Hilbert spaces and , respectively. Consider the generalized product on as in Lemma 8 with width . Assume that is a map the range of which contains all rank-one projections. Then satisfies for all if and only if one of the following conditions holds. (1)There exists a unitary operator such that for all .(2)There exists a unitary operator such that for all ,
where is the transpose of for an arbitrarily but fixed orthonormal basis of .

To prove Theorem 9, we consider the special case by taking and if . By Lemma 8, there exists positive integer with such that . It is clear that Theorem 9 is an immediate consequence of the following result.

Theorem 10. Let and be standard real Jordan algebras of delf-adjoint operators on complex Hilbert spaces and , respectively. Assume that is a map the range of which contains all rank-one projections and is nonnegative integer with . Then satisfies if and only if one of the following conditions holds. (1)There exists a unitary operator such that for all .(2)There exists a unitary operator such that for all ,
where is the transpose of for an arbitrarily but fixed orthonormal basis of .

To prove Theorem 10, it suffices to check the “only if” part. Assume in the following that is a map satisfying (36) with range containing all rank-one projections.

Lemma 11. .

Proof. It follows from (36) that holds for every . Let . For any unit vector , there exists such that . Then which, together with , implies that for all unit vectors . So .

Lemma 12. preserves rank-one projections in both directions.

Proof. For any unit vector , let and . It follows from that . Similarly, we have . If , then , which, together with , implies that , and thus , a contradiction.
So, there exist a unit vector . Take such that . It follows from that , which implies that . Thus and, therefore, preserves rank-one projections.
Similarly one can show that is a rank-one projection will imply that is a rank-one projection.

Proof of Theorem 10. By Lemma 12, preserves rank-one projections in both directions. It follows that there exists a bijective map such that for all unit vectors , where and for any and .
For any unit vectors , we have Hence holds for all . Thus, by Wigner’s theorem again, there exists a unitary or conjugate unitary operator such that for every unit vector .
Now assume that is unitary. Let be arbitrary. For any unit vector , since we have Hence we get for every .
Similarly, is conjugate unitary implies that there exists a unitary operator such that for all .

Remark 13. Finally, we remark that if we do not require that the generalized product is closed in the involved standard real Jordan algebras , , we can still obtain a characterization of the maps from into with range containing all rank-one projections which preserves the peripheral spectrum of an arbitrarily given generalized product. In fact, such maps have the same form stated in Theorem 2.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors wish to give their thanks to the referees for their helpful comments. This work is partially supported by National Natural Science Foundation of China (nos. 11171249 and 11271217).

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Copyright © 2014 Wen Zhang and Jinchuan Hou. 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.

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