Advances in Operator Theory

The matrix power means and interpolations

Trung Hoa Dinh, ‎Raluca Dumitru, and Jose A‎. ‎Franco

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Abstract

‎It is well-known that the Heron mean is a linear interpolation between the arithmetic and the geometric means while the matrix power mean $P_t(A,B):= A^{1/2}\left(\frac{I+(A^{-1/2}BA^{-1/2})^t}{2}\right)^{1/t}A^{1/2}$ interpolates between the harmonic‎, ‎the geometric‎, ‎and the arithmetic means‎. ‎In this article‎, ‎we establish several comparisons between the matrix power mean‎, ‎the Heron mean‎, ‎and the Heinz mean‎. ‎Therefore‎, ‎we have a deeper understanding about the distribution of these matrix means‎.

Article information

Source
Adv. Oper. Theory, Volume 3, Number 3 (2018), 647-654.

Dates
Received: 5 January 2018
Accepted: 28 February 2018
First available in Project Euclid: 4 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.aot/1522807282

Digital Object Identifier
doi:10.15352/aot.1801-1288

Mathematical Reviews number (MathSciNet)
MR3795106

Zentralblatt MATH identifier
06902458

Subjects
Primary: 47A63: Operator inequalities
Secondary: 47A64‎ ‎47A56

Keywords
Kubo-Ando means ‎interpolation‎ ‎arithmetic mean geometric mean harmonic mean ‎Heron means ‎Heinz means ‎power means

Citation

Dinh, Trung Hoa; Dumitru, ‎Raluca; ‎Franco, Jose A‎. The matrix power means and interpolations. Adv. Oper. Theory 3 (2018), no. 3, 647--654. doi:10.15352/aot.1801-1288. https://projecteuclid.org/euclid.aot/1522807282


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References

  • A. Begea, J. Bukor, and J. T. Tóthb, On log-convexity of power mean, Ann. Math. Inform. 42 (2013), 3–7.
  • R. Bhatia, Interpolating the arithmetic-geometric mean inequality and its operator version, Linear Algebra Appl. 413 (2006), no. 2-3, 355–363.
  • R. Bhatia and F. Kittaneh, The matrix arithmetic-geometric mean inequality revisited, Linear Algebra Appl. 428 (2008), no. 8-9, 2177–2191.
  • R. Bhatia, Y. Lim, and T. Yamazaki, Some norm inequalities for matrix means, Linear Algebra Appl. 501 (2016), 112–122.
  • T. H. Dinh, R. Dumitru, and J. A. Franco, On a conjecture of Bhatia, Lim and Yamazaki, Linear Algebra Appl. 532 (2017), 140–145.
  • F. Kubo and T. Ando, Means of positive linear operators, Math. Ann. 246 (1979/80), no. 3, 205–224.
  • Y. Lim and M. Pálfia, Matrix power means and the Karcher mean, J. Funct. Anal. 262 (2012), no. 4, 1498–1514.
  • L. Matejicka, Short note on convexity of power mean, Tamkang J. Math. 46 (2015), no. 4, 423–426.