Kuo, David Li-Wei; Tsai, Ming-Cheng; Wong, Ngai-Ching; Zhang, Jun Maps preserving Schatten \(p\)-norms of convex combinations. (English) Zbl 1473.47014 Abstr. Appl. Anal. 2014, Article ID 520795, 5 p. (2014). Summary: We study maps \(\phi\) of positive operators of the Schatten \(p\)-classes (\(1 < p < + \infty\)), which preserve the \(p\)-norms of convex combinations, that is, \(\| t \rho +(1 - t) \sigma \|_p = \| t \phi(\rho) +(1 - t) \phi(\sigma) \parallel_p\), \(\forall \rho, \sigma \in \mathcal{S}_p^+(H)_1\), \(t \in [0,1]\). They are exactly those carrying the form \(\phi(\rho) = U \rho U^*\) for a unitary or antiunitary \(U\). In the case \(p = 2\), we have the same conclusion whenever it just holds \(\| \rho + \sigma \|_2 = \| \phi(\rho) + \phi(\sigma) \|_2\) for all the positive Hilbert-Schmidt class operators \(\rho, \sigma\) of norm 1. Some examples are demonstrated. Cited in 3 Documents MSC: 47B49 Transformers, preservers (linear operators on spaces of linear operators) 47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) Keywords:maps preserving Schatten \(p\)-norms; positive Hilbert-Schmidt class operators × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Molnár, L., An algebraic approach to Wigner’s unitary-antiunitary theorem, Australian Mathematical Society Journal A, 65, 3, 354-369 (1998) · Zbl 0943.46033 · doi:10.1017/S144678870003593X [2] Chan, J.-T.; Li, C.-K.; Sze, N.-S., Isometries for unitarily invariant norms, Linear Algebra and Its Applications, 399, 53-70 (2005) · Zbl 1073.15022 · doi:10.1016/j.laa.2004.05.017 [3] Li, C.-K.; Pierce, S., Linear preserver problems, The American Mathematical Monthly, 108, 7, 591-605 (2001) · Zbl 0991.15001 · doi:10.2307/2695268 [4] Cassinelli, G.; De Vito, E.; Lahti, P.; Levrero, A., Symmetry groups in quantum mechanics and the theorem of Wigner on the symmetry maps, Reviews in Mathematical Physics, 8, 921-941 (1997) · Zbl 0907.46055 [5] Simon, B.; Lieb, E. H.; Simon, B.; Wightman, A. S., Quantum dynamics: from automorphism to hamiltonian, Studies in Mathematical Physics. Essays in Honor of Valentine Bargmann. Studies in Mathematical Physics. Essays in Honor of Valentine Bargmann, Princeton Series in Physics, 327-349 (1976), Princeton University Press · Zbl 0343.47034 [6] Molnár, L., Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces. Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces, Lecture Notes in Mathematics, 1895 (2007), Berlin, Germany: Springer, Berlin, Germany · Zbl 1119.47001 [7] Nagy, G., Isometries on positive operators of unit norm, Publicationes Mathematicae Debrecen, 82, 183-192 (2013) · Zbl 1299.47072 [8] Abatzoglou, T. J., Norm derivatives on spaces of operators, Mathematische Annalen, 239, 2, 129-135 (1979) · Zbl 0398.47013 · doi:10.1007/BF01420370 [9] McCarthy, C. A., \(c_p\), Israel Journal of Mathematics, 5, 249-271 (1967) · Zbl 0156.37902 · doi:10.1007/BF02771613 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.