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General Preservers of Quasi-Commutativity

Published online by Cambridge University Press:  20 November 2018

Gregor Dolinar  and
Bojan Kuzma
Gregor Dolinar
Affiliation:
Faculty of Electrical Engineering, University of Ljubljana, Ljubljana, Slovenia e-mail: gregor.dolinar@fe.uni-lj.si
Bojan Kuzma
Affiliation:
University of Primorska, Koper, Slovenia and Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia, e-mail: bojan.kuzma@pef.upr.si
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Abstract

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Let ${{M}_{n}}$ be the algebra of all $n\,\times \,n$ matrices over $\mathbb{C}$. We say that $A,B\in {{M}_{n}}$ quasi-commute if there exists a nonzero $\xi \,\in \,\mathbb{C}$ such that $AB\,=\,\xi BA$. In the paper we classify bijective not necessarily linear maps $\Phi :{{M}_{n}}\to {{M}_{n}}$ which preserve quasi-commutativity in both directions.

Keywords

15A0415A2706A99general preserversmatrix algebraquasi-commutativity
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 62 , Issue 4 , 01 August 2010 , pp. 758 - 786
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] Baribeau, L. and Ransford, T., Non-linear spectrum-preserving maps. Bull. London Math. Soc. 32(2000), no. 1, 8–14. doi:10.1112/S0024609399006426Google Scholar
[2] Bhatia, R. and Rosenthal, P., How and why to solve the operator equation AX − XB = Y. Bull. London Math. Soc. 29(1997), no. 1, 1–21. doi:10.1112/S0024609396001828Google Scholar
[3] Brooke, J. A., Busch, P., and Pearson, D. B., Commutativity up to a factor of bounded operators in complex Hilbert space. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458(2002), no. 2017, 109–118. doi:10.1098/rspa.2001.0858Google Scholar
[4] Cassinelli, G., De Vito, E., Lahti, P. J., and Levrero, A., The theory of symmetry actions in quantum mechanics. With an application to the Galilei group. Lecture Notes in Physics, 654, Springer-Verlag, Berlin, 2004.Google Scholar
[5] Chan, J.-T., Li, C.-K., and Sze, N.-S., Mappings on matrices: invariance of functional values of matrix products. J. Aust. Math. Soc. 81(2006), no. 2, 165–184. doi:10.1017/S1446788700015809Google Scholar
[6] Cui, J., Hou, J., and Park, C., Additive maps preserving commutativity up to a factor. Chinese Ann. Math. Ser. A 29(2008), no. 5, 583–590.Google Scholar
[7] Dolinar, G., Du, S., Hou, J., and P. Legiša, General preservers of invariant subspace lattices. Linear Algebra Appl. 429(2008), no. 1, 100–109. doi:10.1016/j.laa.2008.02.007Google Scholar
[8] Du, S., Hou, J., and Bai, Z., Nonlinear maps preserving similarity on B(H). Linear Algebra Appl. 422(2007), no. 2–3, 506–516. doi:10.1016/j.laa.2006.11.008Google Scholar
[9] Fillmore, F. A., Herrero, D. A., and W. Longstaff, E., The hyperinvariant subspace lattice of a linear transformation. Linear Algebra Appl. 17(1977), no. 2, 125–132. doi:10.1016/0024-3795(77)90032-5Google Scholar
[10] Holtz, O., Mehrmann, V., and Schneider, H., Potter, Wielandt, and Drazin on the matrix equation AB = !BA: new answers to old questions. Amer. Math. Monthly 111(2004), no. 8, 655–667. doi:10.2307/4145039Google Scholar
[11] Hoffman, K. and Kunze, R., Linear algebra. Second ed., Prentice-Hall, Englewoods Cliffs, NJ, 1971.Google Scholar
[12] Horn, R. A. and Johnson, C. R., Topics in matrix analysis. Cambridge University Press, Cambridge, 1991.Google Scholar
[13] Hua, L.-K., A theorem on matrices over a sfield and its applications. J. Chinese Math. Soc. (N.S.) 1(1951), 110–163.Google Scholar
[14] Mc Coy, N., On quasi-commutative matrices. Trans. Amer. Math. Soc. 36(1934), no. 2, 327–340. doi:10.2307/1989841Google Scholar
[15] Molnár, L., Orthogonality preserving transformations on indefinite inner product spaces: generalization of Uhlhorn's version of Wigner's theorem. J. Funct. Anal. 194(2002), no. 2, 248–262. doi:10.1006/jfan.2002.3970Google Scholar
[16] Molnár, L., Linear maps on matrices preserving commutativity up to a factor. Linear and Multilinear Algebra 57(2009), no. 1, 13–18. doi:10.1080/03081080701210211Google Scholar
[17] Molnár, L. and Šemrl, P., Nonlinear commutativity preserving maps on self-adjoint operators. Q. J. Math. 56(2005), no. 4, 589–595. doi:10.1093/qmath/hah058Google Scholar
[18] von Neumann, J., Mathematical foundations of quantum mechanics. Princeton University Press, Princeton, NJ, 1955.Google Scholar
[19] Omladič, M., Radjavi, H., and Šemrl, P., Preserving commutativity. J. Pure Appl. Algebra 156(2001), no. 2-3, 309–328. doi:10.1016/S0022-4049(99)00154-1Google Scholar
[20] Ovchinnikov, P. G., Automorphisms of the poset of skew projections. J. Funct. Anal. 115(1993), no. 1, 184–189. doi:10.1006/jfan.1993.1086Google Scholar
[21] Potter, H. S. A., On the latent roots of quasi-commutative matrices. Amer. Math. Monthly 57(1950), 321–322. doi:10.2307/2306202Google Scholar
[22] Radjavi, H. and Šemrl, P., Linear maps preserving quasi-commutativity. Studia Math. 184(2008), no. 2, 191–204. doi:10.4064/sm184-2-7Google Scholar
[23] Šemrl, P., Non-linear commutativity preserving maps. Acta Sci. Math. (Szeged) 71(2005), no. 3–4, 781–819.Google Scholar
[24] Šemrl, P., Maps on idempotent matrices over division rings. J. Algebra 298(2006), no. 1, 142–187. doi:10.1016/j.jalgebra.2005.08.010Google Scholar
[25] Šemrl, P., Maps on matrix spaces. Linear Algebra Appl. 413(2006), no. 2–3, 364–393. doi:10.1016/j.laa.2005.03.011Google Scholar
[26] Šemrl, P., Commutativity preserving maps. Linear Algebra Appl. 429(2008), no. 5–6, 1051–1070. doi:10.1016/j.laa.2007.05.006Google Scholar
[27] Watkins, W., Linear maps that preserve commuting pairs of matrices. Linear Algebra and Appl. 14(1976), no. 1, 29–35. doi:10.1016/0024-3795(76)90060-4Google Scholar
[28] Wedderburn, J. H. M., Lectures on matrices. Dover Publications, New York, 1964.Google Scholar