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Isometries for unitarily invariant norms. (English) Zbl 1073.15022

After a brief survey of results and proof techniques in the study of isometries for unitarily invariant norms on real and complex rectangular matrices, the paper presents a characterization of a class of linear isometries without the linearity assumption. Some related results and problems like the invariant norms on other matrix and operator algebras and spaces, and isometry problems without the surjectivity assumption are finally discussed.

MSC:

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A04 Linear transformations, semilinear transformations
15A18 Eigenvalues, singular values, and eigenvectors
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[1] Anoussis, M.; Katavolos, A., Isometries of Cp-spaces in nest algebras, J. London math. soc., 51, 2, 175-188, (1995) · Zbl 0846.47026
[2] Arazy, J., The isometries of Cp, Israel J. math., 22, 247-256, (1975)
[3] Arazy, J., Isometries of Banach algebras satisfying the von Neumann inequality, Math. scand., 74, 137-151, (1994) · Zbl 0818.46064
[4] Arazy, J.; Friedman, Y., The isometries of \(C_p^{n, m}\) into C, Israel J. math., 26, 151-165, (1977) · Zbl 0345.47035
[5] Arazy, J.; Solel, B., Isometries of nonselfadjoint operator algebras, J. funct. anal., 90, 284-305, (1990) · Zbl 0713.46043
[6] Auerbach, H., Sur LES groupes bornés de substitutions linéaires, C.R. acad. sci. Paris, 195, 1367-1369, (1932) · JFM 58.0128.05
[7] Chan, J.T.; Li, C.K.; Tu, C.C.N., A class of unitarily invariant norms on B(H), Proc. amer. math. soc., 129, 1065-1076, (2001) · Zbl 0963.47013
[8] Chang, S.; Li, C.K., A special linear operator on Mn(R), Linear and multilinear algebra, 30, 65-76, (1991)
[9] Charzyński, Z., Sur LES transformations isométriques des espaces du type (F), Studia math., 13, 94-121, (1953) · Zbl 0051.08504
[10] W.S.Cheung, C.K. Li, Y.T. Poon, Isometries Between Matrix Algebras J. Austral. Math. Soc., in press
[11] Chu, C.-H.; Dang, T.; Russo, B.; Ventura, B., Surjective isometries of real C*-algebras, J. London math. soc., 47, 2, 97-118, (1993) · Zbl 0732.46037
[12] Deutsch, E.; Schneider, H., Bounded groups and norm-Hermitian matrices, Linear algebra appl., 9, 9-27, (1974) · Zbl 0298.15017
[13] Djokovic, D.Z., On isometries of matrix spaces, Linear and multilinear algebra, 27, 73-78, (1990) · Zbl 0706.15015
[14] Djokovic, D.Z.; Li, C.K., Overgroups of some classical linear groups with applications to linear preserver problems, Linear algebra appl., 197/198, 31-62, (1994)
[15] Fan, K.; Hoffman, A.J., Some metric inequalities in the space of matrices, Proc. amer. math. soc., 6, 111-116, (1955) · Zbl 0064.01402
[16] R. Grone, Isometries of matrix algebras, Ph.D. Thesis, U.C. Santa Barbara, 1976 · Zbl 0359.15012
[17] Grone, R., The invariance of partial isometries, Indiana univ. math. J., 28, 445-449, (1979) · Zbl 0379.15008
[18] Grone, R., Certain isometries of rectangular complex matrices, Linear algebra appl., 29, 161-171, (1980) · Zbl 0432.15015
[19] Grone, R.; Marcus, M., Isometries of matrix algebras, J. algebra, 47, 180-189, (1977) · Zbl 0359.15012
[20] Guralnick, R.; Li, C.K., Invertible preservers and algebraic groups III: preservers of unitary similarity (congruence) invariants and overgroups of some unitary subgroups, Linear and multilinear algebra, 43, 257-282, (1997) · Zbl 0889.20027
[21] Horn, R.; Mathias, R., Cauchy-Schwarz inequalities associated with positive semidefinite matrices, Linear algebra appl., 142, 63-82, (1990) · Zbl 0714.15012
[22] Jacobson, N., Some groups of transformations defined by Jordan algebra I, J. reine angew. math., 201, 178-195, (1959) · Zbl 0084.03601
[23] Jacobson, N.; Rickart, C., Jordan homomorphisms of rings, Trans. amer. math. soc., 69, 479-502, (1950) · Zbl 0039.26402
[24] Johnson, C.R.; Laffey, T.; Li, C.K., Linear transformations on \(M_n(\mathbb{R})\) that preserve the Ky Fan k-norm and a remarkable special case when (n,k)=(4,2), Linear and multilinear algebra, 23, 285-298, (1988) · Zbl 0658.15029
[25] Kadison, R.V., Isometries of operator algebras, Ann. math., 54, 2, 325-338, (1951) · Zbl 0045.06201
[26] Kadison, R.V., A generalized Schwarz inequality and algebraic invariants for operator algebras, Ann. of math., 56, 2, 494-503, (1952) · Zbl 0047.35703
[27] C.K. Li, Some results on generalized spectral radii, numerical radii and spectral norms, Ph.D. Thesis, University of Hong Kong, 1986
[28] Li, C.K., Some aspects of the theory of norms, Linear algebra appl., 212/213, 71-100, (1994) · Zbl 0814.15021
[29] C.K. Li, Y.T. Poon, N.S. Sze, Isometries for Ky Fan norms between matrix spaces, Proc. Amer. Math. Soc., in press · Zbl 1067.15027
[30] Li, C.K.; Šemrl, P.; Sourour, A., Isometries for Ky-Fan norms on block matrix algebras, Arch. math. (basel), 81, 175-181, (2003) · Zbl 1048.15003
[31] Li, C.K.; Tsing, N.K., Some isometries of rectangular matrices, Linear and multilinear algebra, 23, 47-53, (1988) · Zbl 0641.15005
[32] Li, C.K.; Tsing, N.K., Duality between some linear preserver problems II: isometries with respect to c-spectral norms and matrices with fixed singular values, Linear algebra appl., 110, 181-212, (1988) · Zbl 0655.15026
[33] Li, C.K.; Tsing, N.K., Linear operators preserving unitarily invariant norms on matrices, Linear and multilinear algebra, 26, 119-132, (1990) · Zbl 0691.15006
[34] Li, C.K.; Tsing, N.K., Duality between some linear preserver problems III: c-spectral norms on (skew-)symmetric matrices and matrices with fixed singular values, Linear algebra appl., 143, 67-97, (1990) · Zbl 0712.15028
[35] Marcus, M., All linear operators leaving the unitary group invariant, Duke math. J., 26, 155-163, (1959) · Zbl 0084.01701
[36] Marcus, M., Linear transformations on matrices, J. res. nat. bur. standards sect. B, 75B, 107-113, (1971) · Zbl 0244.15013
[37] Marcus, M.; Moyls, B., Transformations on tensor product spaces, Pac. J. math., 9, 1215-1221, (1956) · Zbl 0089.08902
[38] McCarthy, C.A., Cp, Israel J. math., 5, 249-271, (1967)
[39] Minc, H., Linear transformations on matrices: rank 1 preservers and determinant preservers, Linear and multilinear algebra, 4, 265-272, (1977) · Zbl 0351.15005
[40] Mirsky, L., Symmetric gauge functions and unitarily invariant norms, Quart. J. math. Oxford, 11, 2, 50-59, (1960) · Zbl 0105.01101
[41] Moore, R.L., Isomorphisms for CSL algebras, Indiana univ. math. J., 52, 687-702, (2003) · Zbl 1051.47055
[42] Moore, R.L.; Trent, T.T., Isometries of nest algebras, J. funct. anal., 86, 180-209, (1989) · Zbl 0693.47036
[43] Moore, R.L.; Trent, T.T., Isometries of certain reflexive operator algebras, J. funct. anal., 98, 437-471, (1991) · Zbl 0743.47034
[44] Omladič, M.; Šemrl, P., Additive mappings preserving operators of rank one, Linear algebra appl., 182, 239-256, (1993) · Zbl 0803.47026
[45] Russo, B., Trace preserving mappings of matrix algebras, Duke math. J., 36, 297-300, (1969) · Zbl 0181.40703
[46] Russo, B., Isometries of the trace class, Proc. amer. math. soc., 23, 213., (1969) · Zbl 0181.40704
[47] I. Schur, Einige Bemerkungen zur Determinantentheorie, Sitzungsberichte der Preuss. Akad. Wiss. zu Berlin 25 (1925) 454-463 · JFM 51.0110.04
[48] Solel, B., Isometries of CSL algebras, Trans. amer. math. soc., 332, 595-606, (1992) · Zbl 0772.47024
[49] Sourour, A., Isometries of norm ideals of compact operators, J. funct. anal., 43, 69-77, (1981) · Zbl 0474.47009
[50] Von Neumann, J., Some matrix-inequalities and metrization of matrix-space, Tomsk. univ. rev., 1, 286-300, (1937) · Zbl 0017.09802
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