Discrepancies in Euclidean Operator Radii in Hilbert C∗-Modules

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DOI: https://doi.org/10.28924/2291-8639-22-2024-174
Published: Oct 2, 2024
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M.H.M. Rashid, Rabaa Al-Maita, Discrepancies in Euclidean Operator Radii in Hilbert C∗-Modules, Int. J. Anal. Appl., 22 (2024), 174.

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M.H.M. Rashid, Rabaa Al-Maita

Abstract

In this research, we establish precise limits for the Euclidean operator radius of two bounded linear operators operating within a Hilbert C∗-module over A. Furthermore, our work establishes a connection between these limits and recent research findings that provide accurate upper and lower bounds for the numerical radius of linear operators. The primary objective of this investigation is to explore various specific scenarios of interest and extend existing inequalities found in the literature to encompass the Euclidean radius of two operators in a Hilbert A-module. Additionally, our study presents conclusions that reveal relationships between the operator norm, the typical numerical radius of a composite operator, and the Euclidean operator radius. Furthermore, we introduce several new inequalities involving the Euclidean numerical radius and Euclidean operator norm of 2-tuple operators. These inequalities offer both lower and upper bounds for the Euclidean numerical radius of 2-tuple operators, as well as for the sum and product of 2-tuple operators. We also delve into the study of Euclidean numerical radius inequalities for 2×2 operator matrices whose entries consist of 2-tuple operators, leading to the derivation of some Euclidean operator radius inequalities. Additionally, we establish an inequality for the Euclidean operator norm of 2×2 operator matrices.

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References

  1. M. Cho, M. Takaguchi, Boundary Points of Joint Numerical Ranges, Pac. J. Math. 95 (1981), 27–35. https://doi.org/10.2140/pjm.1981.95.27.
  2. S. S. Dragomir, Some Refinements of Schwartz Inequality, in: Proceedings of the Symposium of Mathematics and Its Applications, Timisoara Research Centre of the Romanian Academy, Timisoara, 1986, pp. 13–16.
  3. P.R. Halmos, A Hilbert Space Problem Book, Springer, New York, 1982.
  4. G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, Cambridge University Press, Cambridge, 1988.
  5. R. Jiang, A Note on the Triangle Inequality for the C∗-Valued Norm on a Hilbert C∗-Module, Math. Ineq. Appl. 3 (2013), 743–749. https://doi.org/10.7153/mia-16-56.
  6. I. Kaplansky, Modules over Operator Algebras, Amer. J. Math. 75 (1953), 839–858. https://doi.org/10.2307/2372552.
  7. F. Kittaneh, Notes on Some Inequalities for Hilbert Space Operators, Publ. Res. Inst. Math. Sci. 24 (1988), 276–293. https://doi.org/10.2977/prims/1195175202.
  8. F. Kittaneh, Norm Inequalities for Certain Operator Sums, J. Funct. Anal. 143 (1997), 337–348. https://doi.org/10.1006/jfan.1996.2957.
  9. R. Kaur, M.S. Moslehian, M. Singh, C. Conde, Further Refinements of the Heinz Inequality, Linear Algebra Appl. 447 (2014), 26–37. https://doi.org/10.1016/j.laa.2013.01.012.
  10. E.C. Lance, Hilbert C∗-Module: A Toolkit for Operator Algebraists, London Mathematical Society Lecture Note Series 210, Cambridge University Press, Cambridge, 1995.
  11. M. Mehrazin, M. Amyari, M.E. Omidvar, A New Type of Numerical Radius of Operators on Hilbert C∗-Module, Rend. Circ. Mat. Palermo, II. Ser. 69 (2018), 29–37. https://doi.org/10.1007/s12215-018-0385-3.
  12. S.F. Moghaddam, A.K. Mirmostafaee, Numerical Radius Inequalities for Hilbert C∗-Modules, Math. Bohem. 147 (2022), 547–566. http://eudml.org/doc/298846.
  13. W.L. Paschke, Inner Product Modules Over B∗-Algebras, Trans. Amer. Math. Soc. 182 (1973), 443–468. https://doi.org/10.1090/s0002-9947-1973-0355613-0.
  14. G. Popescu, Unitary Invariants in Multivariable Operator Theory, American Mathematical Society, Providence, 2009.
  15. J.E. Pecaric, T. Furuta, Y. Seo, Mond-Pecaric Method in Operator Inequalities: Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005.
  16. C. Pearcy, An Elementary Proof of the Power Inequality for the Numerical Radius, Michigan Math. J. 13 (1966), 289–291.
  17. M.H.M. Rashid, Some Inequalities for the Euclidean Operator Radius of Two Operators in Hilbert C∗-Modules Space, arXiv:2307.01695, (2023). https://doi.org/10.48550/ARXIV.2307.01695.
  18. M.M.H. Rashid, Some Inequalities for the Numerical Radius and Spectral Norm for Operators in Hilbert C∗-Modules Space, Tamkang J. Math. (2024). https://doi.org/10.5556/j.tkjm.56.2025.5167.
  19. M.H.M. Rashid, W.M.M. Salameh, Inequalities for the Euclidean Operator Radius of n-Tuple Operators and Operator Matrices in Hilbert C∗-Modules, Symmetry 16 (2024), 647. https://doi.org/10.3390/sym16060647.
  20. M.A. Rieffel, Induced Representations of C∗-Algebras. Adv. Math. 13 (1974), 176–257.
  21. N.E. Wegge-Olsen, K-Theory and C∗-Algebras: A Friendly Approach, Oxford University Press, Oxford, 1993.