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Norm inequalities related to Clarkson inequalities. (English) Zbl 1390.15064
Summary: Let \(A\) and \(B\) be \(n\times n\) matrices. It is shown that if \(p=2\), \(4\leq p<\infty\), or \(2<p<4\), and both \(A+B\), \(A-B\) are positive semidefinite, then \[ \|A+B\|^p_p + \|A-B\|^p_p \leq 2^{p-1} \left ( \|A\|^p_p + \|B\|^p_p \right )-\left (2^{p/2}-2\right )\big| \|A\|_p -\|B\|_p\big|^p, \] and if \(p=2\), \(4 \leq p < \infty\), or \(2<p<4\), and both \(A\), \(B\) are positive semidefinite, then \[ \|A+B\|^p_p + \|A-B\|^p_p \geq 2 \left ( \|A\|^p_p + \|B\|^p_p \right )+(2^{1-p/2}-2^{2-p})\big| \|A+B\|_p- \|A-B\|_p\big|^p. \] These inequalities are reversed if \(p=2\), \(1 \leq p \leq \frac43\), or \(\frac43 <p<2\), and both \(A+B\), \(A-B\) are positive semidefinite, and if \(p=2\), \(1 \leq p \leq \frac43\), or \(\frac43 < p < 2\), and both \(A\), \(B\) are positive semidefinite, respectively. Commutative (or \(L_p\)) versions of these inequalities are also considered.

MSC:
15A45 Miscellaneous inequalities involving matrices
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
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.)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
15A18 Eigenvalues, singular values, and eigenvectors
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References:
[1] K.M.R. Audenaert and F. Kittaneh. Problems and conjectures in matrix and operator inequalities. Banach Center Publ., 112:15-31, 2017. · Zbl 1381.15008
[2] K. Ball, E. Carlen, and E.H. Lieb. Sharp uniform convexity and smoothness inequalities for trace norms. Invent. Math., 115:463-482, 1994. · Zbl 0803.47037
[3] R. Bhatia and J. Holbrook. On the Clarkson-McCarthy inequalities. Math. Ann., 281:7-12, 1988. · Zbl 0618.47008
[4] R. Bhatia and F. Kittaneh. Clarkson inequalities with several operators. Bull. Lond. Math. Soc., 36:820-832, 2004. · Zbl 1071.47011
[5] E. Carlen and E.H. Lieb. Some Matrix rearrangement inequalities. Ann. Mat. Pura Appl., 185:315-324, 2006. · Zbl 1197.15011
[6] J.A. Clarkson. Uniform Convex Spaces. Trans. Amer. Math. Soc., 40:396-414, 1936. · JFM 62.0460.04
[7] C. Conde and M.S. Moslehian. Norm inequalities related to p-Schatten class. Linear Algebra Appl., 498:441-449, 2016. · Zbl 1341.47011
[8] T. Fack and H. Kosaki. Generalized s-numbers of τ -measurable operators. Pacific J. Math., 123:269-300, 1986. · Zbl 0617.46063
[9] O. Hanner. On the uniform convexity of L p and l p. Ark. Mat., 3:239-244, 1956. · Zbl 0071.32801
[10] O. Hirzallah and F. Kittaneh. Non-commutative Clarkson inequalities for unitarily invariant norms. Pacific J. Math., 202:363-369, 2002. · Zbl 1054.47011
[11] O. Hirzallah and F. Kittaneh.Non-commutative Clarkson inequalities for n-tuples of operators.Integral Equations Operator Theory, 60:369-379, 2008. · Zbl 1155.47013
[12] E. Kissin. On Clarkson-McCarthy inequalities for n-tuples of operators. Proc. Amer. Math. Soc., 135:2483-2495, 2007. · Zbl 1140.47005
[13] B. Simon. Trace Ideals and their Applications. Cambridge University Press, Cambridge, 1979. · Zbl 0423.47001
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