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DISCUSSIONS ON PARTIAL ISOMETRIES IN BANACH SPACES AND BANACH ALGEBRAS

  • Received : 2016.01.20
  • Published : 2017.03.31

Abstract

The aim of this paper is twofold. Firstly, we introduce the concept of semi-partial isometry in a Banach algebra and carry out a comparison and a classification study for this concept. In particular, we show that in the context of $C^*$-algebras this concept coincides with the notion of partial isometry. Our results encompass several earlier ones concerning partial isometries in Hilbert spaces, Banach spaces and $C^*$-algebras. Finally, we study the notion of (m, p)-semi partial isometries.

References

  1. J. Agler and M. Stankus, m-isometric transformations of Hilbert space. I, Integral Equa-tions and Operator Theory 21 (1995), no. 4, 383-429. https://doi.org/10.1007/BF01222016
  2. J. Agler and M. Stankus, m-isometric transformations of Hilbert space. II, Integral Equations and Oper-ator Theory 23 (1995), no. 1, 1-48. https://doi.org/10.1007/BF01261201
  3. J. Agler and M. Stankus, m-isometric transformations of Hilbert space. III, Integral Equations and Op-erator Theory 24 (1996), no. 4, 379-421. https://doi.org/10.1007/BF01191619
  4. F. Bayart, m-isometries on Banach spaces, Math. Nachr. 284 (2011), no. 17-18, 2141-2147. https://doi.org/10.1002/mana.200910029
  5. E. Boasso, On the Moore-Penrose inverse, EP Banach space operators, and EP Banach algebra elements, J. Math. Anal. Appl. 339 (2008), no. 2, 1003-1014. https://doi.org/10.1016/j.jmaa.2007.07.059
  6. F. F. Bonsall and J. Duncan, Numerical Ranges of Operators on Normed Spaces and Elements of Normed Algebras, Cambridge Univ. Press, London, 1971.
  7. A. A. Bourhim, M. Burgos, and V. S. Shulman, Linear maps preserving the minimum and reduced minimum moduli, J. Funct. Anal. 285 (2010), no. 1, 50-66.
  8. A. Bourhim and M. Mabrouk, Numerical radius and product of elements in C*-algebras, Linear Multilinear Algebra 65 (2017), no. 6, 1108-1116. https://doi.org/10.1080/03081087.2016.1228818
  9. A. Browder, On Bernstein's inequality and the norm of hermitian operators, Amer. Math. Monthly 78 (1971), no. 8, 871-873. https://doi.org/10.2307/2316478
  10. M. Fernandez-Miranda and J. PH. Labrousse, Moore-penrose inverses and finite range elements in a C*-algebra, Rev. Roumaine Math. Pures Appl. 45 (2000), 609-630.
  11. T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1996.
  12. M. Mbekhta, Conorme et inverse generalise dans les C*-algebres, Canad. Math. Bull. 35 (1995), no. 4, 515-522. https://doi.org/10.4153/CMB-1992-068-8
  13. M. Mbekhta, Partial isometries and generalized inverses, Acta. Sci. Math. (Szeged) 70 (2004), no. 3-4, 767-781.
  14. D. Mosic and D. S. Djordjevic, Partial isometries and EP elements in Banach algebras, Abstr. Appl. Anal. 2011 (2011), Art. ID 540212, 9 pp.
  15. R. Penrose, A generalized inverse for matrices, Mathematical proceedings of the Cam-bridge philosophical society, vol. 51, pp. 406-413, Cambridge Univ Press, 1955.
  16. C. Schmoeger, On a question of Mbekhta, Extracta Math. 20 (2005), no. 3, 281-290.
  17. C. Schmoeger, Generalized projections in Banach algebras, Linear Algebra Appl. 430 (2009), no. 2-3, 601-608. https://doi.org/10.1016/j.laa.2008.07.020
  18. X. H. Sun and Y. Li, The reduced minimum modulus of left multiplicative operators, J. Shandong Univ. Nat. Sci. 45 (2010), no. 2, 54-57.
  19. M. A. Taoudi, On a generalization of partial isometries on Banach spaces, Georgian Math. J. 15 (2008), no. 1, 177-188.
  20. Y. Xue, Stable perturbations of operators and related topics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012.