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Maps preserving operator pairs whose products are projections. (English) Zbl 1206.47032

Let \(H\) be a complex Hilbert space with \(\dim H \geq 2\) and let \(B(H)\) be the algebra of all bounded linear operators on \(H\). The authors show that, if \(\phi: B(H)\to B(H)\) is a surjective map preserving operator pairs whose products are nonzero projections in both directions (i.e., \(\phi (A) \phi (B)\) is a nonzero projection if and only if \(AB\) is), then there exists a unitary or an anti-unitary operator \(U\) on \(H\) and a constant \(\lambda\) satisfying \(\lambda^2 =1\) such that \(\phi (T)= \lambda U^* T U\) for all \(T \in B(H)\). In [Acta Math.Sin., Chin.Ser.53, No.2, 315–322 (2010; Zbl 1206.47033)], the authors considered linear maps satisfying the same property.
Now let \(A,B \in B(H)\). Recall that the triple Jordan product of \(A\) and \(B\) is \(ABA\). Analogously, the authors show that, if \(\phi: B(H)\to B(H)\) is a surjective map preserving operator pairs whose triple Jordan products are nonzero projections in both directions, then there exists a unitary or an anti-unitary operator \(U\) on \(H\) and a constant \(\lambda\) satisfying \(\lambda^3 =1\) such that either \(\phi (T)= \lambda U^* T U\) for all \(T \in B(H)\) or \(\phi (T)= \lambda U^* T^* U\) for all \(T \in B(H)\).

MSC:

47B49 Transformers, preservers (linear operators on spaces of linear operators)

Citations:

Zbl 1206.47033
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References:

[1] Botta, P.; Pierce, S.; Watkins, W., Linear transformations that preserve the nilpotent matrices, Pacfic J. Math., 104, 39-46 (1983) · Zbl 0446.15002
[2] Brešar, M.; Šemrl, P., On locally linearly dependent operators and derivation, Trans. Amer. Math. Soc., 351, 1257-1275 (1999) · Zbl 0920.15009
[3] Clark, S.; Li, C.-K.; Rodman, L., Spectral radius preservers of products of nonnegative matrices, Banach. J. Math. Annal, 2, 107-120 (2008) · Zbl 1158.15002
[4] Dobovišek, M.; Kuzma, B.; Lešnjak, G.; Li, C. K.; Petek, T., Mappings that preserve pairs of operators with zero triple Jordan product, Linear Algebra Appl., 426, 255-279 (2007) · Zbl 1130.15002
[5] Fang, L.; Ji, G.; Pang, Y., Maps preserving the idempotency of products of operators, Linear Algebra Appl., 426, 40-52 (2007) · Zbl 1131.47036
[6] Gudder, S.; Nagy, G., Sequential quantum measurements, J. Math. Phys., 42, 5212-5222 (2001) · Zbl 1018.81005
[7] Halmos, P. R., A Hilbert Space Problem Book (1982), Springer-Verlag: Springer-Verlag New York · Zbl 0144.38704
[8] Hou, J.; Di, Q., Maps preserving numerical ranges of operator products, Proc. Amer. Math. Soc., 134, 1435-1446 (2006) · Zbl 1089.47030
[9] Hou, J.; Li, C.-K.; Wong, N.-C., Jordan isomorphisms and maps preserving spectra of certain operator products, Studia. Math., 184, 31-47 (2008) · Zbl 1134.47028
[10] Hou, J.; Li, C.-K.; Wong, N.-C., Maps preserving the spectrum of generalized Jordan products of operators, Linear Algebra Appl., 432, 1049-1069 (2010) · Zbl 1185.47036
[11] Ji, G.; Qu, F., Linear maps preserving projections of products of operators, Acta Math. Sinica, 53, 2, 315-322 (2010) · Zbl 1206.47033
[12] Li, C.-K.; Šemrl, P.; Sze, N.-S., Maps preserving the nilpotency of products of operators, Linear Algebra Appl., 424, 222-239 (2007) · Zbl 1124.47026
[13] Molnár, L., Maps on states preserving the relative entropy, J. Math. Phys., 49, 032114 (2008) · Zbl 1153.81407
[14] Molnár, L.; Šemrl, P., Nonlinear commutativity preserving maps on self-adjoint operators, Quart. J. Math., 56, 589-595 (2005) · Zbl 1211.47075
[15] Šemrl, P., Linear maps that preserve the nilpotent operators, Acta Sci. Math. (Szeged), 61, 523-534 (1995) · Zbl 0843.47024
[16] Šemrl, P., Non-linear commutativity preserving maps, Acta Sci. Math. (Szeged), 71, 781-819 (2005) · Zbl 1111.15002
[17] P. Šemrl, Maps on idempotent operators, Perspectives in operator theory, Banach Center Publ., 75, Polish Acad. Sci., Warsaw, 2007, pp. 289-301.; P. Šemrl, Maps on idempotent operators, Perspectives in operator theory, Banach Center Publ., 75, Polish Acad. Sci., Warsaw, 2007, pp. 289-301.
[18] Uhlhorn, U., Representation of symmetry transformations in quantum mechanics, Ark Fysik, 23, 307-340 (1963) · Zbl 0108.21805
[19] Wigner, E. P., Group Theory and its Application to the Quantum Theory of Atomic Spectra (1959), Academic Press Inc.: Academic Press Inc. New York · Zbl 0085.37905
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