Clarkson inequalities related to convex and concave functions

Fugen Gao; Meng Li

doi:10.1515/ms-2021-0055

Article

Clarkson inequalities related to convex and concave functions

Fugen Gao and Meng Li

Published/Copyright: October 4, 2021

Published by

Become an author with De Gruyter Brill

Author Information Explore this Subject

From the journal Mathematica Slovaca Volume 71 Issue 5

Abstract

In this paper, we obtain some norm inequalities involving convex and concave functions, which are the generalizations of the classical Clarkson inequalities. Let A₁, …, A_n be bounded linear operators on a complex separable Hilbert space H and let α₁, …, α_n be positive real numbers such that ∑j=1nαj=1. We show that for every unitarily invariant norm,

If f is a non-negative function on [0, ∞) such that f(0) = 0 and g(t)=f(t) is convex, then
|||∑j=1nαjf(|Aj|)|||≥|||∑j,k∈Sℓ(f(αjαk4αℓ(1−αℓ)|Aj+Ak−2∑j=1nαjAj|)+f(αjαk(2αℓ−1)4αℓ(1−αℓ)|Aj−Ak|))+f(|∑j=1nαjAj|)|||
for ℓ = 1, …, n.
If f is a non-negative function on [0, ∞) such that g(t)=f(t) is concave, then the inverse inequality holds. Here, the symbol S_ℓ = {1, …, n} ∖ {ℓ} for ℓ ∈ {1, …, n}.

In addition, we provide some applications of the above inequalities.

Keywords: Clarkson inequalities; unitarily invariant norm; Schatten p-norm; convex function; concave function

MSC 2010: Primary 47A30; Secondary 47B10; 47B15; 46B20

This research is supported by the National Natural Science Foundation of China (11601339, 11701154), the Natural Science Foundation of the Department of Education, Henan Province (19A110020, 20A110020), the graduate education reform and quality improvement project, Henan province and higher education teaching reform and practice project (postgraduate education) of Henan Normal University (YJS2019JG01).

(Communicated by Emanuel Chetcuti)

References

[1] Alrimawi, F.—Hirzallah, O.—Kittaneh, F.: Norm inequalities involving convex and concave functions of operators, Linear Multilinear Algebra 67 (2019), 1757–1772.10.1080/03081087.2018.1470601Search in Google Scholar

[2] Aujla, J.—Silva, F.: Weak majorization inequalities and convex functions, Linear Algebra Appl. 369 (2003), 217–233.10.1016/S0024-3795(02)00720-6Search in Google Scholar

[3] Bhatia, R.: Matrix Analysis, Springer-Verlag, New York, 1997.10.1007/978-1-4612-0653-8Search in Google Scholar

[4] Bhatia, R.—Holbrook, J.: On the Clarkson-McCarthy inequalities, Math. Ann. 281 (1988), 7–12.10.1007/BF01449211Search in Google Scholar

[5] Bhatia, R.—Kittaneh, F.: Clarkson inequalities with several operators, Bull. London Math. Soc. 36 (2004), 820–832.10.1112/S0024609304003467Search in Google Scholar

[6] Bourin, J. C.—Uchiyama, M.: A matrix subadditivity inequality for f(A + B) and f(A) + f(B), Linear Algebra Appl. 423 (2007), 512–518.10.1016/j.laa.2007.02.019Search in Google Scholar

[7] Fack, T.—Kosaki, H.: Generalized s-numbers of τ-measurable operators, Pacific J. Math. 123 (1986), 269–300.10.2140/pjm.1986.123.269Search in Google Scholar

[8] Formisano, T.—Kissin, E.: Clarkson-McCarthy inequalities for ℓ_p spaces of operators in Schatten ideals, Integral Equ. Oper. Theory 79 (2014), 151–173.10.1007/s00020-014-2145-xSearch in Google Scholar

[9] Gohberg, I. C.—Krein, M. G.: Introduction to the Theory of Linear Nonselfadjoint Operators. Transl. Math. Monographs 18, Amer. Math. Soc. Providence, RI, 1969.Search in Google Scholar

[10] Gumus, I. H.—Hirzallah, O.—Kittaneh, F.: Eigenvalue localization for complex matrices, Electron. J. Linear Algebra 27 (2014), 892–906.10.13001/1081-3810.2866Search in Google Scholar

[11] Gumus, I. H.—Hirzallah, O.—Kittaneh, F.: Estimates for the real and imaginary parts of the eigenvalues of matrices and applications, Linear Multilinear Algebra 36 (2016), 2431–2445.10.1080/03081087.2016.1160997Search in Google Scholar

[12] Hirzallah, O.—Kittaneh, F.: Non-commutative Clarkson inequalities for unitarily invariant norms, Pacific J. Math. 202 (2002), 363–369.10.2140/pjm.2002.202.363Search in Google Scholar

[13] Hirzallah, O.—Kittaneh, F.: Non-commutative Clarkson inequalities for n-tuples of operator, Integral Equ. Oper. Theory 60(3) (2008), 369–379.10.1007/s00020-008-1565-xSearch in Google Scholar

[14] Kissin, E.: On Clarkson-McCarthy inequalities for n-tuples of operators, Proc. Amer. Math. Soc. 135 (2007), 2483–2495.10.1090/S0002-9939-07-08710-2Search in Google Scholar

[15] Kosem, T.: Inequalities between ∣f(A + B)∣ and ∣f(A) + f(B)∣, Linear Algebra Appl. 418 (2006), 153–160.10.1016/j.laa.2006.01.028Search in Google Scholar

[16] McCarthy, C. A.: c_p, Isr. J. Math. 5 (1967), 249–271.10.1007/BF02771613Search in Google Scholar

[17] Simon, B.: Trace Ideals and their Applications, Cambridge University Press, Cambridge, 1979.Search in Google Scholar

[18] Uchiyama, M.: Subadditivity of eigenvalue sums, Proc. Amer. Math. Soc. 134 (2006), 1405–1412.10.1090/S0002-9939-05-08116-5Search in Google Scholar

Received: 2021-03-18

Accepted: 2021-06-18

Published Online: 2021-10-04

Published in Print: 2021-10-26

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/ms-2021-0055

Keywords for this article

Clarkson inequalities; unitarily invariant norm; Schatten p-norm; convex function; concave function