Continuous generalization of Clarkson-McCarthy inequalities. (English) Zbl 07002030

Summary: Let \(G\) be a compact Abelian group, let \(\mu\) be the corresponding Haar measure, and let \(\hat{G}\) be the Pontryagin dual of \(G\). Furthermore, let \(\mathcal{C}_{p}\) denote the Schatten class of operators on some separable infinite-dimensional Hilbert space, and let \(L^{p}(G;\mathcal{C}_{p})\) denote the corresponding Bochner space. If \(G\ni\theta\mapsto A_{\theta}\) is the mapping belonging to \(L^{p}(G;\mathcal{C}_{p})\), then \[ \begin{aligned}\sum_{k\in\hat{G}}\| \int_{G}\overline{k(\theta)}A_{\theta}\,\text{d}\theta\|_{p}^{p}&\leq\int_{G}\| A_{\theta}\|_{p}^{p}\,\text{d}\theta,\quad p\geq2,\\ \sum_{k\in\hat{G}}\| \int_{G}\overline{k(\theta)}A_{\theta}\,\text{d}\theta\|_{p}^{p}&\leq(\int_{G}\| A_{\theta}\|_{p}^{q}\,\text{d}\theta )^{p/q},\quad p\geq2,\\ \sum_{k\in\hat{G}}\| \int_{G}\overline{k(\theta)}A_{\theta}\,\text{d}\theta\|_{p}^{q}&\leq(\int_{G}\| A_{\theta}\|_{p}^{p}\,\text{d}\theta )^{q/p},\quad p\leq2.\end{aligned} \] If \(G\) is a finite group, then the previous equations comprise several generalizations of Clarkson-McCarthy inequalities obtained earlier (e.g., \(G=\mathbf{Z}_{n}\) or \(G=\mathbf{Z}_{2}^{n}\)), as well as the original inequalities, for \(G=\mathbf{Z}_{2}\). We also obtain other related inequalities.


47A30 Norms (inequalities, more than one norm, etc.) of linear operators
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
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[1] K. M. R. Audenaert and J. S. Aujla, On norm sub-additivity and super-additivity inequalities for concave and convex functions, Linear Multlinear Algebra 60 (2012), no. 11-12, 1369-1389. · Zbl 1260.15031
[2] J. S. Aujla and F. C. Silva, Weak majorization inequalities and convex functions, Linear Algebra Appl. 369 (2003), 217-233. · Zbl 1031.47007
[3] R. Bhatia and F. Kittaneh, Clarkson inequalities with several operators, Bull. Lond. Math. Soc. 36 (2004), no. 6, 820-832. · Zbl 1071.47011
[4] R. P. Boas, Some uniformly convex spaces, Bull. Amer. Math. Soc. (N.S.) 46 (1940), no. 4, 304-311. · JFM 66.0533.03
[5] J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40, no. 3 (1936), 396-414. · Zbl 0015.35604
[6] R. E. Edwards, Integration and Harmonic Analysis on Compact Groups, London Math. Soc. Lecture Note Ser. 8, Cambridge Univ. Press, London, 1972. · Zbl 0231.43001
[7] T. Fack and H. Kosaki, Generalized \(s\)-numbers of \(τ\)-measurable operators, Pacific J Math. 123 (1986), no. 2, 269-300. · Zbl 0617.46063
[8] G. B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., Textb. Math., CRC Press, Boca Raton, FL., 2016. · Zbl 1342.43001
[9] T. Formisano and E. Kissin, Clarkson-McCarthy inequalities for \(lp\)-spaces of operators in Shatten ideals, Integral Equations Operator Theory 79 (2014), no. 2, 151-173. · Zbl 1310.46016
[10] I. C. Gohberg and M. G. Kreĭn, Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monogr. 18, Amer. Math. Soc., Providence, 1969. · Zbl 0181.13504
[11] K. Hashimoto, M. Kato, and Y. Takahashi, Generalized Clarkson’s inequalities for Lebesgue-Bochner spaces, Bull. Kyushu Inst. Technol. Pure Appl. Math. 43 (1996), 15-25. · Zbl 0874.46021
[12] O. Hirzallah and F. Kittaneh, Non-commutative Clarkson inequalities for \(n\)-tuples of operators, Integral Equations Operator Theory 60 (2008), no. 3, 369-379. · Zbl 1155.47013
[13] J. Karamata, Sur une inégalité relative aux fonctions convexes, Publ. Inst. Math. (Beograd) (N.S.) 1 (1932), 145-148. · Zbl 0005.20101
[14] M. Kato, Generalized Clarkson’s inequalities and the norms of the Littlewood matrices, Math. Nachr. 114 (1983), no. 1, 163-170. · Zbl 0578.47018
[15] E. Kissin, On Clarkson-McCarthy inequalities for \(n\)-tuples of operators, Proc. Amer. Math. Soc. 135 (2007), no. 8, 2483-2495. · Zbl 1140.47005
[16] T. Kosem, Inequalities between \(\Vert f({A}+{B})\Vert\) and \(\Vert f({A})+f({B})\Vert \), Linear Algebra Appl. 418 (2006), no. 1, 153-160. · Zbl 1105.15016
[17] M. Koskela, Some generalizations of Clarkson’s inequalities, Univ. Beograd. Publ. Eletrotehn. Fak. Ser. Mat. Fiz. 634-677 (1979), 89-93. · Zbl 0477.46029
[18] L. Maligranda and L. E. Persson, On Clarkson’s inequalities and interpolation, Math. Nachr. 155 (1992), 187-197. · Zbl 0777.46041
[19] C. A. McCarthy, \({C}_p\), Israel J. Math. 5 (1967), 249-271.
[20] B. Simon, Trace Ideals and Their Applications, London Math. Soc. Lecture Note Ser. 35, Cambridge Univ. Press, Cambridge, 1979. · Zbl 0423.47001
[21] M. Uchiyama, Subadditivity of eigenvalue sums, Proc. Amer. Math. Soc. 134 (2006), no. 5, 1405-1412. · Zbl 1089.47010
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