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On Clarkson–McCarthy inequalities for $$n$$-tuples of operators. (English) Zbl 1140.47005
Noncommutative analogues of Clarkson inequalities for pairs of operators in Schatten ideals obtained by McCarthy play an important role in analysis and operator theory. They were generalized for general symmetric norms by various authors, e.g., Bhatia, Hirzallah, Holbrook, Kittaneh. In [R. Bhatia and F. Kittaneh, Bull. Lond. Math. Soc., 36, 820–832 (2004; Zbl 1071.47011)], analogues of Clarkson–McCarthy inequalities were proved for $$n$$-tuples of operators of special type. In the paper under review, the author obtains analogues of Clarkson–McCarthy inequalities for all $$n$$-tuples of operators. As applications, he extends some inequalities for partitioned operators and for Cartesian decompositions of operators.

##### MSC:
 47A30 Norms (inequalities, more than one norm, etc.) of linear operators 46B20 Geometry and structure of normed linear spaces 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.) 47A13 Several-variable operator theory (spectral, Fredholm, etc.)
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##### References:
 [1] Tsuyoshi Ando and Xingzhi Zhan, Norm inequalities related to operator monotone functions, Math. Ann. 315 (1999), no. 4, 771 – 780. · Zbl 0941.47004 · doi:10.1007/s002080050335 · doi.org [2] Keith Ball, Eric A. Carlen, and Elliott H. Lieb, Sharp uniform convexity and smoothness inequalities for trace norms, Invent. Math. 115 (1994), no. 3, 463 – 482. · Zbl 0803.47037 · doi:10.1007/BF01231769 · doi.org [3] Rajendra Bhatia, Matrix analysis, Graduate Texts in Mathematics, vol. 169, Springer-Verlag, New York, 1997. · Zbl 0863.15001 [4] Rajendra Bhatia and John A. R. Holbrook, On the Clarkson-McCarthy inequalities, Math. Ann. 281 (1988), no. 1, 7 – 12. · Zbl 0618.47008 · doi:10.1007/BF01449211 · doi.org [5] Rajendra Bhatia and Fuad Kittaneh, Norm inequalities for partitioned operators and an application, Math. Ann. 287 (1990), no. 4, 719 – 726. · Zbl 0688.47005 · doi:10.1007/BF01446925 · doi.org [6] Rajendra Bhatia and Fuad Kittaneh, Cartesian decompositions and Schatten norms, Linear Algebra Appl. 318 (2000), no. 1-3, 109 – 116. · Zbl 0981.47008 · doi:10.1016/S0024-3795(00)00206-8 · doi.org [7] Rajendra Bhatia and Fuad Kittaneh, Clarkson inequalities with several operators, Bull. London Math. Soc. 36 (2004), no. 6, 820 – 832. · Zbl 1071.47011 · doi:10.1112/S0024609304003467 · doi.org [8] Введение в теорию линейных несамосопряженных операторов в гил$$^{\приме}$$бертовом пространстве, Издат. ”Наука”, Мосцощ, 1965 (Руссиан). · Zbl 0138.07803 [9] Omar Hirzallah and Fuad Kittaneh, Non-commutative Clarkson inequalities for unitarily invariant norms, Pacific J. Math. 202 (2002), no. 2, 363 – 369. · Zbl 1054.47011 · doi:10.2140/pjm.2002.202.363 · doi.org [10] Charles A. McCarthy, \?_\?, Israel J. Math. 5 (1967), 249 – 271. · Zbl 0156.37902 · doi:10.1007/BF02771613 · doi.org [11] Barry Simon, Trace ideals and their applications, London Mathematical Society Lecture Note Series, vol. 35, Cambridge University Press, Cambridge-New York, 1979. · Zbl 0423.47001
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