Journal of the Mathematical Society of Japan

Convergence of Aluthge iteration in semisimple Lie groups

Tin-Yau TAM and Mary Clair THOMPSON

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Abstract

We extend the $\lambda$-Aluthge sequence convergence theorem of Antezana, Pujals and Stojanoff in the context of real noncompact connected semisimple Lie groups.

Article information

Source
J. Math. Soc. Japan, Volume 66, Number 4 (2014), 1127-1131.

Dates
First available in Project Euclid: 23 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1414090237

Digital Object Identifier
doi:10.2969/jmsj/06641127

Mathematical Reviews number (MathSciNet)
MR3272594

Zentralblatt MATH identifier
1305.22012

Subjects
Primary: 22E46: Semisimple Lie groups and their representations

Keywords
Aluthge transform Aluthge iteration semisimple Lie group Cartan decomposition

Citation

TAM, Tin-Yau; THOMPSON, Mary Clair. Convergence of Aluthge iteration in semisimple Lie groups. J. Math. Soc. Japan 66 (2014), no. 4, 1127--1131. doi:10.2969/jmsj/06641127. https://projecteuclid.org/euclid.jmsj/1414090237


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References

  • J. Antezana, P. Massey and D. Stojanoff, $\lambda$-Aluthge transforms and Schatten ideals, Linear Algebra Appl., 405 (2005), 177–199.
  • J. Antezana, E. Pujals and D. Stojanoff, Iterated Aluthge transforms: a brief survey, Rev. Un. Mat. Argentina, 49 (2008), 29–41.
  • J. Antezana, E. Pujals and D. Stojanoff, Convergence of the iterated Aluthge transform sequence for diagonalizable matrices. II, $\lambda$-Aluthge transform, Integral Equations Operator Theory, 62 (2008), 465–488.
  • J. Antezana, E. Pujals and D. Stojanoff, Convergence of the iterated Aluthge transform sequence for diagonalizable matrices, Adv. Math., 216 (2007), 255–278.
  • J. Antezana, E. Pujals and D. Stojanoff, The iterated Aluthge transforms of a matrix converge, Adv. Math., 226 (2011), 1591–1620.
  • A. Aluthge, On $p$-hyponormal operators for $0<p<1$, Integral Equations Operator Theory, 13 (1990), 307–315.
  • A. Aluthge, Some generalized theorems on $p$-hyponormal operators, Integral Equations Operator Theory, 24 (1996), 497–501.
  • T. Ando and T. Yamazaki, The iterated Aluthge transforms of a 2-by-2 matrix converge, Linear Algebra Appl., 375 (2003), 299–309.
  • M. Chō, I. B. Jung and W. Y. Lee, On Aluthge transforms of $p$-hyponormal operators, Integral Equations Operator Theory, 53 (2005), 321–329.
  • C. Foiaş, I. B. Jung, E. Ko and C. Pearcy, Complete contractivity of maps associated with the Aluthge and Duggal transforms, Pacific J. Math., 209 (2003), 249–259.
  • S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure Appl. Math., 80, Academic Press, New York, 1978.
  • H. Huang and T.-Y. Tam, On the convergence of Aluthge sequence, Oper. Matrices, 1 (2007), 121–141.
  • H. Huang and T.-Y. Tam, Aluthge iteration in semisimple Lie group, Linear Algebra Appl., 432 (2010), 3250–3257.
  • A. W. Knapp, Lie Groups Beyond an Introduction, Progr. Math., 140, Birkhäuser, Boston, 1996.
  • A. A. Sagle and R. E. Walde, Introduction to Lie Groups and Lie Algebras, Pure Appl. Math. (Amst.), 51, Academic Press, New York, 1973.
  • T.-Y. Tam, $\lambda$-Aluthge iteration and spectral radius, Integral Equations Operator Theory, 60 (2008), 591–596.
  • T. Takasaki, Basic indexes and Aluthge transformation for 2 by 2 matrices, Math. Inequal. Appl., 11 (2008), 615–628.
  • D. Wang, Heinz and McIntosh inequalities, Aluthge transformation and the spectral radius, Math. Inequal. Appl., 6 (2003), 121–124.
  • T. Yamazaki, An expression of spectral radius via Aluthge transformation, Proc. Amer. Math. Soc., 130 (2002), 1131–1137.