## Annals of Functional Analysis

### On the Araki–Lieb–Thirring inequality in the semifinite von Neumann algebra

Yazhou Han

#### Abstract

This paper extends a recent matrix trace inequality of Bourin–Lee to semifinite von Neumann algebras. This provides a generalization of the Lieb–Thirring-type inequality in von Neumann algebras due to Kosaki. Some new inequalities, even in the matrix case, are also given for the Heinz means.

#### Article information

Source
Ann. Funct. Anal., Volume 7, Number 4 (2016), 622-635.

Dates
Received: 15 December 2015
Accepted: 5 May 2016
First available in Project Euclid: 23 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.afa/1474652186

Digital Object Identifier
doi:10.1215/20088752-3660864

Mathematical Reviews number (MathSciNet)
MR3550940

Zentralblatt MATH identifier
06667758

Subjects
Primary: 47A63: Operator inequalities
Secondary: 46L52: Noncommutative function spaces

#### Citation

Han, Yazhou. On the Araki–Lieb–Thirring inequality in the semifinite von Neumann algebra. Ann. Funct. Anal. 7 (2016), no. 4, 622--635. doi:10.1215/20088752-3660864. https://projecteuclid.org/euclid.afa/1474652186

#### References

• [1] M. Alakhrass, Inequalities related to Heinz mean, Linear Multilinear Algebra 64 (2015), no. 8, 1562–1569.
• [2] H. Araki, On an inequality of Lieb and Thirring, Lett. Math. Phys. 19 (1990), no. 2, 167–170.
• [3] T. N. Bekjan and D. Dauitbek, Submajorization inequalities of $\tau$-measurable operators for concave and convex functions, Positivity 19 (2015), no. 2, 341–345.
• [4] T. N. Bekjan and G. Massen, Submajorization of the Araki–Lieb–Thirring inequality, Kyushu J. Math. 69 (2015), no. 2, 387–392.
• [5] A. M. Bikchentaev, On normal $\tau$-measurable operators affliliated with semifinite von Neumann algebras, Math. Notes 96 (2014), no. 3–4, 332–341.
• [6] J. C. Bourin and E. Y. Lee, Matrix inequalities from a two variables functional, preprint, arXiv:1511.06977.
• [7] P. G. Dodds and T. K. Dodds, Some aspects of the theory of symmetric operator spaces, Quaest. Math. 18 (1995), no. 1–3, 47–89.
• [8] P. G. Dodds, T. K. Dodds, P. N. Dowling, C. J. Lennard, and F. A. Sukochev, A uniform Kadec-Klee property for symmetric operator spaces, Math. Proc. Canib. Phil. Soc. 118 (1995), no. 3, 487–502.
• [9] P. G. Dodds, T. K. Dodds, and F. A. Sukochev, On p-convexity and q-concavity in noncommutative symmetric spaces, Integral Equations Operator Theory 78 (2014), no. 1, 91–114.
• [10] P. G. Dodds and F. A. Sukochev, Submajorisation inequalities for convex and concave functions of sums of measurable operators, Positivity 13 (2009), no. 1, 107–124.
• [11] T. Fack, Sur la notion de valeur caracteristique, J. Operator Theory 7 (1982), no. 2, 307–333.
• [12] T. Fack and H. Kosaki, Generalized s-numbers of $\tau$-measurable operators, Pac. J. Math. 123 (1986), no. 2, 269–300.
• [13] F. Hansen, An operator inequality, Math. Ann. 246 (1980), no. 3, 249–250.
• [14] F. Hiai, A generalization of Araki’s log-majorization, Linear Algebra Appl. 501 (2016), 1–16.
• [15] H. Kosaki, Applications of uniform convexity of noncommutative $L^{p}$-spaces, Trans. Amer. Math. Soc. 283 (1984), no. 1, 265–282.
• [16] H. Kosaki, An inequality of Araki–Lieb–Thirring (von Neumann algebra case), Proc. Amer. Math. Soc. 114 (1992), no. 2, 477–481.
• [17] E. Lieb and W. Thirring, “Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities” in Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann, Princeton Univ. Press, Princeton, 1976, 269–303.
• [18] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, II: Function Spaces, Ergeb. Math. Grenzgeb. 97, Springer, Berlin, 1979.
• [19] Y. Nakamura, An inequality for generalized s-numbers, Integral Equations Operator Theory 10 (1987), no. 1, 140–145.
• [20] G. Pisier and Q. Xu, “Noncommutative $L^{p}$-spaces” in Handbook of the Geometry of Banach Spaces, Vol. 2, North-Holland, Amsterdam, 2003, 1459–1517.
• [21] M. Terp, $L^{p}$ spaces associated with von Neumann algebras, lecture notes, Univ. of Copenhagen, Copenhagen, 1981.
• [22] Q. Xu, Analytic functions with values in lattices and symmetric spaces of measurable operators, Math. Proc. Camb. Phil. Soc. 109 (1991), no. 3, 541–563.