Elements of C*-algebras commuting with their Moore-Penrose inverse

J. Koliha

Studia Mathematica (2000)

  • Volume: 139, Issue: 1, page 81-90
  • ISSN: 0039-3223

Abstract

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We give new necessary and sufficient conditions for an element of a C*-algebra to commute with its Moore-Penrose inverse. We then study conditions which ensure that this property is preserved under multiplication. As a special case of our results we recover a recent theorem of Hartwig and Katz on EP matrices.

How to cite

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Koliha, J.. "Elements of C*-algebras commuting with their Moore-Penrose inverse." Studia Mathematica 139.1 (2000): 81-90. <http://eudml.org/doc/216712>.

@article{Koliha2000,
abstract = {We give new necessary and sufficient conditions for an element of a C*-algebra to commute with its Moore-Penrose inverse. We then study conditions which ensure that this property is preserved under multiplication. As a special case of our results we recover a recent theorem of Hartwig and Katz on EP matrices.},
author = {Koliha, J.},
journal = {Studia Mathematica},
keywords = {C*-algebra; Moore-Penrose inverse; Drazin inverse; -algebra; regular elements; EP matrices},
language = {eng},
number = {1},
pages = {81-90},
title = {Elements of C*-algebras commuting with their Moore-Penrose inverse},
url = {http://eudml.org/doc/216712},
volume = {139},
year = {2000},
}

TY - JOUR
AU - Koliha, J.
TI - Elements of C*-algebras commuting with their Moore-Penrose inverse
JO - Studia Mathematica
PY - 2000
VL - 139
IS - 1
SP - 81
EP - 90
AB - We give new necessary and sufficient conditions for an element of a C*-algebra to commute with its Moore-Penrose inverse. We then study conditions which ensure that this property is preserved under multiplication. As a special case of our results we recover a recent theorem of Hartwig and Katz on EP matrices.
LA - eng
KW - C*-algebra; Moore-Penrose inverse; Drazin inverse; -algebra; regular elements; EP matrices
UR - http://eudml.org/doc/216712
ER -

References

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  1. [1] T. S. Baskett and I. J. Katz, Theorems on products of E P r matrices, Linear Algebra Appl. 2 (1969), 87-103. Zbl0179.05104
  2. [2] K. G. Brock, A note on commutativity of a linear operator and its Moore-Penrose inverse, Numer. Funct. Anal. Optim. 11 (1990), 673-678. Zbl0729.47001
  3. [3] S. L. Campbell and C. D. Meyer, Generalized Inverses of Linear Transformations, Pitman, London, 1979. Zbl0417.15002
  4. [4] M. P. Drazin, Pseudo-inverse in associative rings and semigroups, Amer. Math. Monthly 65 (1958), 506-514. Zbl0083.02901
  5. [5] R. E. Harte and M. Mbekhta, On generalized inverses in C*-algebras, Studia Math. 103 (1992), 71-77. Zbl0810.46062
  6. [6] R. E. Harte and M. Mbekhta, Generalized inverses in C*-algebras II, ibid. 106 (1993), 129-138. Zbl0810.46063
  7. [7] R. Hartwig and I. J. Katz, On products of EP matrics, Linear Algebra Appl. 252 (1997), 339-345. Zbl0868.15015
  8. [8] I. J. Katz, Weigman type theorems for E P r matrices, Duke Math. J. 32 (1965), 423-428. 
  9. [9] J. J. Koliha, A generalized Drazin inverse, Glasgow Math. J. 38 (1996), 367-381. Zbl0897.47002
  10. [10] J. J. Koliha, The Drazin and Moore-Penrose inverse in C*-algebras, Proc. Roy. Irish Acad. Sect. A 99 (1999), 17-27. Zbl0943.46031
  11. [11] J. J. Koliha, A simple proof of the product theorem for EP matrices, Linear Algebra Appl. 294 (1999), 213-215. Zbl0938.15017
  12. [12] I. Marek and K. Žitný, Matrix Analysis for Applied Sciences, Vol. 2, Teubner, Leipzig, 1986. Zbl0613.15002
  13. [13] R. Penrose, A generalized inverse for matrices, Proc. Cambridge Philos. Soc. 51 (1955), 406-413. Zbl0065.24603
  14. [14] E. T. Wong, Does the generalized inverse of A commute with A?, Math. Mag. 59 (1986), 230-232. Zbl0611.15007
  15. [15] D. Djordjević, Products of EP operators on Hilbert spaces, Proc. Amer. Math. Soc., to appear. 
  16. [16] G. Lešnjak, Semigroups of EP linear transformations, Linear Algebra Appl. 304 (2000), 109-118. Zbl0946.15009

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