7
This article was downloaded by: [Ams/Girona*barri Lib] On: 21 November 2014, At: 03:01 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Linear and Multilinear Algebra Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/glma20 Linear maps on matrices preserving commutativity up to a factor Lajos Molnár a a Institute of Mathematics , University of Debrecen , 4010 Debrecen, PO Box 12, Hungary Published online: 23 Oct 2008. To cite this article: Lajos Molnár (2009) Linear maps on matrices preserving commutativity up to a factor, Linear and Multilinear Algebra, 57:1, 13-18, DOI: 10.1080/03081080701210211 To link to this article: http://dx.doi.org/10.1080/03081080701210211 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms- and-conditions

Linear maps on matrices preserving commutativity up to a factor

  • Upload
    lajos

  • View
    214

  • Download
    1

Embed Size (px)

Citation preview

Page 1: Linear maps on matrices preserving commutativity up to a factor

This article was downloaded by: [Ams/Girona*barri Lib]On: 21 November 2014, At: 03:01Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Linear and Multilinear AlgebraPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/glma20

Linear maps on matrices preservingcommutativity up to a factorLajos Molnár aa Institute of Mathematics , University of Debrecen , 4010Debrecen, PO Box 12, HungaryPublished online: 23 Oct 2008.

To cite this article: Lajos Molnár (2009) Linear maps on matrices preserving commutativity up to afactor, Linear and Multilinear Algebra, 57:1, 13-18, DOI: 10.1080/03081080701210211

To link to this article: http://dx.doi.org/10.1080/03081080701210211

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever orhowsoever caused arising directly or indirectly in connection with, in relation to or arisingout of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Page 2: Linear maps on matrices preserving commutativity up to a factor

Linear and Multilinear Algebra, Vol. 57, No. 1, January 2009, 13–18

Linear maps on matrices preserving

commutativity up to a factor

LAJOS MOLNAR*

Institute of Mathematics, University of Debrecen, 4010 Debrecen, PO Box 12, Hungary

Communicated by P. Semrl

(Received 16 July 2006; revised 19 November 2006; in final form 27 December 2006)

In this note we determine the general form of all bijective linear maps on the algebra of n� ncomplex matrices or on the space of n� n self-adjoint matrices which preserve the relation ofcommutativity up to a factor in both directions.

Keywords: Preservers; Matrix spaces; Commutativity up to a factor

1991 Mathematics Subject Classifications: Primary; 15A04; 15A27

Linear preserver problems deal with the characterizations of linear maps on matrixspaces which leave invariant a given quantity attached to matrices, or a given relationamong matrices, or a given subset of matrices, etc. These problems represent one of themost extensively investigated research areas in matrix theory over the past severaldecades. A lot of effort has been done also to generalizations or extensions of suchproblems in different directions. Here we mention the study of linear preservers oninfinite dimensional structures (operator algebras, function algebras, etc.) and thestudy of such preservers which are not supposed to be linear (additive preservers,multiplicative preservers and, recently, preservers which are not supposed to respectany algebraic operation).

Problems concerning commutativity preservers belong to the group of the mostextensively investigated preserver problems. From their large literature here we onlyrefer to the classical paper of Watkins [6] and a recent important paper of Bresarand Semrl [1]. Commutativity has serious applications in the mathematicalfoundations of quantum mechanics. One of the most important relations among

*Email: [email protected]

Linear and Multilinear AlgebraISSN 0308-1087 print/ISSN 1563-5139 online � 2009 Taylor & Francis

www.informaworld.comDOI: 10.1080/03081080701210211

Dow

nloa

ded

by [

Am

s/G

iron

a*ba

rri L

ib]

at 0

3:01

21

Nov

embe

r 20

14

Page 3: Linear maps on matrices preserving commutativity up to a factor

quantum observables is compatibility which means that the operators representing theobservables in question commute.

In their paper Brooke et al. [2] have pointed out another relation between operatorswhich is closely related to commutativity and showed its applications in quantummechanics. This is called commutativity up to a factor. If A,B are Hilbert spaceoperators, then they are said to commute up to a factor if there exists a nonzeroscalar �2C such that AB ¼ �BA. In [2] and [5], the authors explored conditions onpairs of operators under which they are in such a relation. The possible values of thefactor � were formulated and shown to depend on the spectral properties of the opera-tors A,B. Several important results concerning the relation under consideration in thematrix case were presented in the paper [4].

The aim of this note is to consider and solve a linear preserver problem for therelation of commutativity up to a factor. The study of questions of that kind wasinitiated by Zhihao Ma.

In what follows we prove that any bijective linear map on the space of all n� n com-plex matrices which preserves the above defined relation in both directions is necessarilya constant multiple of either an algebra automorphism or an algebra antiautomorphismof the underlying matrix algebra. It should be emphasised that the situation with thecorresponding commutativity preservers is different. In fact, in their general form alinear functional multiplied by the identity also appears as the relation of commutativitybetween operators and is clearly preserved under the addition of scalar operators.Moreover, we point out that our result describes the preservers under considerationon every matrix spaceMn(C) with n� 2, so the case n¼ 2 is not exceptional as with com-mutativity preservers [6]. In the last part of the article we determine our preservers alsoon the space of all self-adjoint matrices.

To formulate the main result we introduce the following notation: for any pairA,B 2MnðCÞ of matrices we write A�f B if they commute up to a factor.

THEOREM Let 2 � n 2 N. Suppose that � : MnðCÞ !MnðCÞ is a bijective linear mapwhich has the property that

A �f B()�ðAÞ �f �ðBÞ

holds for any pair A,B 2MnðCÞ of matrices. Then we have a nonzero scalar c 2 C and anonsingular matrix T 2MnðCÞ such that � is either of the form

�ðAÞ ¼ cTAT�1 ðA2MnðCÞÞ

or of the form

�ðAÞ ¼ cTAtT�1 ðA2MnðCÞÞ:

Here, At denotes the transpose of A.

The proof of the theorem rests on the following observation.

LEMMA Let 0 6¼ A 2MnðCÞ. The set of all matrices in MnðCÞ which commute with A upto a factor forms a linear subspace of MnðCÞ if and only if there exist an invertible matrix

14 L. Molnar

Dow

nloa

ded

by [

Am

s/G

iron

a*ba

rri L

ib]

at 0

3:01

21

Nov

embe

r 20

14

Page 4: Linear maps on matrices preserving commutativity up to a factor

S 2MnðCÞ, an integer r, 1 � r � n, and an r� r invertible matrix J with exactly one eigen-value such that A ¼ SðJ� 0ÞS�1. Here, 0 stands for the ðn� rÞ � ðn� rÞ zero matrix.Moreover, in this case for any B 2MnðCÞ we have A�f B if and only if AB¼BA.

Proof First observe that if there is a nonzero scalar � 6¼ 1 and a matrix B 2MnðCÞ

such that

AB ¼ �BA =2 span fAg,

then the set of all matrices which commute with A up to a factor is not a linear space.Indeed, both B and I commute with A up to a factor, but clearly, A �= f ðIþ BÞ.

The second easy observation is that if A ¼ A1 � A2, A1 2MrðCÞ,A2 2Mn�rðCÞ and ifthe set of all matrices in Mn�rðCÞ which commute with A2 up to a factor is not a linearspace, then the set of all matrices in Mn(C) which commute with A up to a factor is alsonot a linear space.

Next, we prove that if A has at least two nonzero eigenvalues, then the set of allmatrices commuting with A up to a factor is not a linear space. Let � 6¼ � be nonzeroeigenvalues of A. Then we can find nonzero n� 1 matrices x and y such that Ax ¼ �xand Aty ¼ �y. It follows that AðxytÞ ¼ �=�ðxytÞA 6¼ spanfAg. So, we are done with ourfirst observation.

We now show that if A is a nonzero nilpotent, then the set of all matrices commutingwith A up to a factor is not a linear space. Assume with no loss of generality that A is amatrix whose entries are all zero except those on the first upper diagonal which areeither 1 or 0. Assume first that the rank of A is at least two. Choose B to be the diagonalmatrix B ¼ diagð2, 4, . . . , 2nÞ. Then we have AB ¼ 2BA =2 spanfAg, and, again, we aredone with our first observation. If A is of rank one, then we may assume thatA ¼ E12. The matrices B ¼ E11 þ E22 and C ¼ E11 � E22 both commute with A up toa factor, while BþC does not.

By the second observation above and the previous paragraph it remains to considerthe case when there exist an invertible matrix S 2MnðCÞ and an r� r invertible matrixJ, 1 � r � n with exactly one eigenvalue ! such that A ¼ SðJ� 0ÞS�1. Here, 0 stands forthe ðn� rÞ � ðn� rÞ zero matrix. With no loss of generality we may assume that S¼ I.Suppose further that a nonzero matrix

B ¼B11 B12

B21 B22

� �

satisfies AB ¼ �BA for some nonzero � 6¼ 1. We get immediately that B12 ¼ B21 ¼ 0 andthat B22 is an arbitrary matrix. If B11¼ 0, then A and B commute. So, we may assumethat A¼ J is an invertible matrix with exactly one eigenvalue ! 6¼ 0 and B 6¼ 0. Takethe Jordan form of A and consider the first nonzero column bi in B. Comparing thei-th columns in AB and �BA we have Abi ¼ �!bi. This gives us that �! 6¼ ! is aneigenvalue of A which is a contradiction. Thus we have proved that in our last casethe set of all matrices commuting with A up to a scalar is equal to A0, the linearspace of all matrices commuting with A. The proof of the lemma is complete. g

Proof of Theorem LetW denote the set of all matrices characterized in Lemma, that is,the set of all matrices which are up to similarity equal either to an invertible matrix with

Preserving commutativity up to a factor 15

Dow

nloa

ded

by [

Am

s/G

iron

a*ba

rri L

ib]

at 0

3:01

21

Nov

embe

r 20

14

Page 5: Linear maps on matrices preserving commutativity up to a factor

exactly one eigenvalue, or to the direct sum of an invertible matrix with exactly oneeigenvalue and the zero matrix. The subset of W consisting of the matrices of theformer type is denoted by W1, while the subset of W formed by the matrices of thelatter type is denoted by W0.

Clearly, we have �ðWÞ ¼W and for every A 2W we also have �ðA0 Þ ¼ �ðAÞ0.Furthermore, for A 2W, the equality A0 ¼MnðCÞ holds true if and only if A is a non-zero scalar matrix. Thus, after multiplying � by a nonzero constant we may assume that�ðI Þ ¼ I.

As the elements A of the set W1 can be characterized among the elements of W withthe property Aþ �I 2W holds for every scalar � but one, it follows that �ðW0Þ ¼W0.It is easy to verify that an element of W0 has maximal commutant exactly when it is anonzero scalar multiple of a nontrivial idempotent (nontriviality means that the idem-potent in question is different from 0 and I ). Hence for every nontrivial idempotentP 2MnðCÞ we have �ðPÞ ¼ cQ for some nonzero scalar c and some nontrivialidempotent Q. But �ðI� PÞ ¼ I� cQ must be a nonzero scalar multiple of a nontrivialidempotent as well. Thus, we have c¼ 1, and consequently, � maps idempotents intoidempotents. It is well-known that every such nonzero linear map on MnðCÞ is eitheran automorphism, or an antiautomorphism. This completes the proof of the theorem.g

Since in quantum mechanics the space of self-adjoint operators plays a distinguishedrole, it is natural to consider our preserver problem also for the space HnðCÞ of all n� nHermitian matrices. The corresponding result reads as follows.

PROPOSITION Let n� 2. Suppose that � : HnðCÞ ! HnðCÞ is a bijective linear map whichpreserves the relation of commutativity up to a factor in both directions. Then there exist anonzero real scalar c and a unitary matrix U 2Mn such that � is either of the form

�ðAÞ ¼ cUAU� ðA 2 HnðCÞÞ

or of the form

�ðAÞ ¼ cUAtU� ðA 2 HnðCÞÞ:

Proof It follows from [2] that commutativity up to a factor between self-adjoint opera-tors is a very special phenomenon: any such pair A,B is either commuting (AB¼BA) oranticommuting (AB ¼ �BA). Now, consider the identity matrix I. It follows that �(I )either commutes or anticommutes with any matrix in HnðCÞ. This implies that �(I )commutes with the squares of the elements of HnðCÞ which means that �(I ) commuteswith every positive semi-definite matrix. Hence, we deduce that �(I) is a scalar matrix.Therefore, there is no loss of generality in assuming that �ðI Þ ¼ I. Pick an elementT 2 HnðCÞ. Every matrix A 2 HnðCÞ which commutes with T is mapped into thematrix �(A) which either commutes or anticommutes with �(T ). Clearly, both kindsof elements of HnðCÞ form a linear subspace and the union of those subspaces is thesubspace of all elements of HnðCÞ which commute with T. As the union of two linearsubspaces is a linear subspace only if one of them contains the other, it follows thatwe have two possibilities:

(i) for every matrix A 2 HnðCÞ which commutes with T we have that �(A) commuteswith �(T),

16 L. Molnar

Dow

nloa

ded

by [

Am

s/G

iron

a*ba

rri L

ib]

at 0

3:01

21

Nov

embe

r 20

14

Page 6: Linear maps on matrices preserving commutativity up to a factor

(ii) for every matrix A 2 HnðCÞ which commutes with T we have that �(A) anticom-mutes with �(T ).

Suppose that T 6¼ 0 and we have the second possibility. As I commutes with T, weobtain that I ¼ �ðIÞ anticommutes which �(T ). But this implies that �ðT Þ ¼ 0 andthen we have T¼ 0, a contradiction. It follows that � preserves commutativity.In fact, as ��1 has the same preserver properties as � we have that � preservescommutativity in both directions. The result [3, Theorem 2] describes the generalform of all commutativity preserving linear transformations on HnðCÞ in the casewhen n� 3. So suppose that n� 3. It follows from the earlier mentioned theorem thatthere exist a nonzero real scalar c, a unitary matrix U 2Mn and a linear functionalf on HnðCÞ such that � is either of the form

�ðAÞ ¼ cUAU� þ f ðAÞI ðA 2 HnðCÞÞ

or of the form

�ðAÞ ¼ cUAtU� þ fðAÞI ðA 2 HnðCÞÞ:

Composing � with appropriate transformations, it is easy to see that without seriousloss of generality we may assume that � is in fact of the form

�ðAÞ ¼ Aþ fðAÞI ðA 2 HnðCÞÞ

with some linear functional f on HnðCÞ. Considering the matrices

A ¼1 00 �1

� �� 0

and

B ¼0 11 0

� �� 0

(where 0 in the direct sums stands for a zero matrix of appropriate size) we see thatA and B anticommute. Therefore, the matrices �(A) and �(B) must either commuteor anticommute. Using elementary matrix computations, one can check that thematrices Aþ f ðAÞI, Bþ fðBÞI cannot commute and they anticommute only iff ðAÞ ¼ fðBÞ ¼ 0. It now implies that �ðAÞ ¼ A. In that way we obtain that � mapsevery rank-two trace zero matrix into itself. Since the system of all those matricestogether with the identity linearly generate HnðCÞ and �ðIÞ ¼ I also holds, we deducethat � is the identity on the whole space HnðCÞ. This completes the proof in the casewhen n� 3.

To avoid boring the reader too much, in the remaining case n¼ 2 we only sketchthe proof. So just as mentioned earlier, we know that � preserves commutativity inboth directions and without loss of generality we can assume that �ðIÞ ¼ I. It is notdifficult to check that the rank-one elements in H2ðCÞ can be characterized in the

Preserving commutativity up to a factor 17

Dow

nloa

ded

by [

Am

s/G

iron

a*ba

rri L

ib]

at 0

3:01

21

Nov

embe

r 20

14

Page 7: Linear maps on matrices preserving commutativity up to a factor

following way: a nonzero and nonscalar element A of H2ðCÞ has rank one if and onlyif the set of all elements which either commute or anticommute with A equals the com-mutant of A. It then follows that � preserves the rank-one elements in H2ðCÞ. Pick twoorthogonal projections P,Q of rank one. We see that �(P) and �(Q) are nonzero scalarmultiples of rank-one projections which commute. As �ðPÞ þ �ðQÞ ¼ �ðIÞ ¼ I, it followsthat those scalars must be 1. This gives us that � preserves the rank-one projections andhence all projections in H2ðCÞ. It is a folk result that every such linear map is a Jordanautomorphism. The structure of those transformations is well-known in any dimension.In fact, we obtain that � is either of the form

�ðAÞ ¼ UAU� ðA 2 H2ðCÞÞ

or of the form

�ðAÞ ¼ UAtU� ðA 2 H2ðCÞÞ

with some unitary U 2MnðCÞ. This completes the proof of the proposition in thecase n¼ 2. g

Acknowledgements

The author is very grateful to the referee for detecting an error in the proof of theoriginal version of Lemma, for correcting it, and for presenting a proof of Theoremsubstantially shorter than the one given in the first version of the article. The authorwas supported by the Hungarian National Foundation for Scientific Research(OTKA), Grant No. T043080, T046203 and by a joint Hungarian-Slovene grant(Reg. No. SLO-5/05).

References

[1] Bresar, M. and Semrl, P., 2005, Commutativity preserving maps on central simple algebras. Journal ofAlgebra, 284, 102–110.

[2] Brooke, J.A., Busch, P. and Pearson, D.B., 2002, Commutativity up to a factor of bounded operators incomplex Hilbert space. Proceedings of the Royal Society of London, Series A Mathematical, Physical andEngineering Sciences, 458, 109–118.

[3] Choi, M.D., Jafarian, A.A. and Radjavi, H., 1987, Linear maps preserving commutativity. Linear Algebraand its Applications, 87, 227–241.

[4] Holtz, O., Mehrmann, V. and Schneider, H., 2004, Potter, Wielandt, and Drazion on the matrix equationAB ¼ !BA: new answers to old questions. American Mathematical Monthly 111, 655–667.

[5] Yang, J. and Du, H.K., 2004, A note on commutativity up to a factor of bounded operators. Proceedingsof American Mathematical Society, 132, 1713–1720.

[6] Watkins, W., 1976, Linear maps that preserve commuting pairs of matrices. Linear Algebra and itsApplications, 14, 29–35.

18 L. Molnar

Dow

nloa

ded

by [

Am

s/G

iron

a*ba

rri L

ib]

at 0

3:01

21

Nov

embe

r 20

14