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Generalized A-Numerical Radius of Operators and Related Inequalities

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Abstract

Let \(\mathcal {H}\) be a complex Hilbert space with inner product \(\langle \cdot , \cdot \rangle \) and let A be a non-zero bounded positive linear operator on \(\mathcal {H}.\) Let \(\mathbb {B}_A(\mathcal {H})\) denote the algebra of all bounded linear operators on \(\mathcal {H}\) which admit A-adjoint, and let \(N_A(\cdot )\) be a seminorm on \(\mathbb {B}_A(\mathcal {H})\). The generalized A-numerical radius of \(T\in \mathbb {B}_A(\mathcal {H})\) is defined as

$$\begin{aligned} \omega _{N_A}(T)=\displaystyle {\sup _{\theta \in \mathbb {R}}}\; N_A\left( \frac{e^{i\theta }T+e^{-i\theta }T^{\sharp _A}}{2}\right) , \end{aligned}$$

where \(T^{\sharp _A}\) stands for a distinguished A-adjoint of T. In this article, we focus on the development of several generalized A-numerical radius inequalities. We also develop bounds for the generalized A-numerical radius of sum and product of operators.

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References

  1. Abu-Omar, A., Kittaneh, F.: A generalization of the numerical radius. Linear Algebra Appl. 569, 323–334 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arias, M.L., Corach, G., Gonzalez, M.C.: Partial isometries in semi-Hilbertian spaces. Linear Algebra Appl. 428(7), 1460–1475 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arias, M.L., Corach, G., Gonzalez, M.C.: Metric properties of projections in semi-Hilbertian spaces. Integral Equ. Oper. Theory 62, 11–28 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arias, M.L., Corach, G., Gonzalez, M.C.: Lifting properties in operator ranges. Acta Sci. Math. (Szeged) 75(3–4), 635–653 (2009)

    MathSciNet  MATH  Google Scholar 

  5. Baklouti, H., Feki, K., Sid Ahmed, O.A.M.: Joint numerical ranges of operators in semi-Hilbertian spaces. Linear Algebra Appl. 555, 266–284 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bani-Domi, W., Kittaneh, F.: Norm and numerical radius inequalities for Hilbert space operators. Linear Multilinear Algebra 69(5), 934–945 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bhunia, P., Paul, K.: New upper bounds for the numerical radius of Hilbert space operators. Bull. Sci. Math. 167, 102959 (2021). https://doi.org/10.1016/j.bulsci.2021.102959

    Article  MathSciNet  MATH  Google Scholar 

  8. Bhunia, P., Feki, K., Paul, K.: Numerical radius inequalities for products and sums of semi-Hilbertian space operators. Filomat 36(4), 1415–1431 (2022)

    Article  MathSciNet  Google Scholar 

  9. Bhunia, P., Nayak, R.K., Paul, K.: Improvement of \(A\)-numerical radius inequalities of semi-Hilbertian space operators. Results Math. 76(3), 120 (2021). https://doi.org/10.1007/s00025-021-01439-w

    Article  MathSciNet  MATH  Google Scholar 

  10. Bhunia, P., Nayak, R.K., Paul, K.: Refinements of \(A\)-numerical radius inequalities and their applications. Adv. Oper. Theory 5, 1498–1511 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bhunia, P., Paul, K., Nayak, R.K.: On inequalities for \(A\)-numerical radius of operators. Electron. J. Linear Algebra 36, 143–157 (2020)

    MathSciNet  MATH  Google Scholar 

  12. Bhunia, P., Sen, A., Paul, K.: New semi-norm of semi-Hilbertian space operators and its application. J. Convex Anal. 29(4) (2022) (to appear)

  13. Bottazzi, T., Conde, C.: Generalized numerical radius and related inequalities. Oper. Matrices 15(4), 1289–1308 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  14. Branges, L.D., Rovnyak, J.: Square Summable Power Series. Holt, Rinehart and Winston, New York (1966)

    MATH  Google Scholar 

  15. Douglas, R.G.: On majorization, factorization and range inclusion of operators in Hilbert space. Proc. Am. Math. Soc. 17, 413–416 (1966)

    Article  MATH  Google Scholar 

  16. Enderami, S.M., Abtahi, M., Zamani, A.: An extension of Birkhoff–James orthogonality relations in semi-Hilbertian space operators. Mediterr. J. Math. (2022) (to appear)

  17. Faghih-Ahmadi, M., Gorjizadeh, F.: \(A\)-numerical radius of \(A\)-normal operators in semi-Hilbertian spaces. Ital. J. Pure Appl. Math. 36, 73–78 (2016)

    MathSciNet  MATH  Google Scholar 

  18. Feki, K.: Spectral radius of semi-Hilbertian space operators and its applications. Ann. Funct. Anal. 11, 929–946 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  19. Feki, K.: Some numerical radius inequalities for semi-Hilbertian space operators. J. Korean Math. Soc. 58(6), 1385–1405 (2021)

    MathSciNet  MATH  Google Scholar 

  20. Feki, K.: A note on the \(A\)-numerical radius of operators in semi-Hilbert spaces. Arch. Math. (Basel) 115(5), 535–544 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  21. Feki, K., Kittaneh, F.: Some new refinements of generalized numerical radius inequalities for Hilbert space operators. Mediterr. J. Math. 19, 17 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  22. Feki, K., Ahmed Mahmoud, S.A.O.: Davis–Wielandt shells of semi-Hilbertian space operators and its applications. Banach J. Math. Anal. 14(3), 1281–1304 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  23. Fong, C.K., Holbrook, J.A.R.: Unitarily invariant operators norms. Can. J. Math. 35, 274–299 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  24. Goldberg, M., Tadmor, E.: On the numerical radius and its applications. Linear Algebra Appl. 42, 263–284 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  25. Halmos, P.: Introduction to Hilbert space and the theory of spectral multiplicity. Chelsea (1951)

  26. Kittaneh, F.: Numerical radius inequalities for Hilbert space operators. Studia Math. 168(1), 73–80 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kittaneh, F., Sahoo, S.: On \(A\)-numerical radius equalities and inequalities for certain operator matrices. Ann. Funct. Anal. 12, 52 (2021). https://doi.org/10.1007/s43034-021-00137-6

    Article  MathSciNet  MATH  Google Scholar 

  28. Saddi, A.: \(A\)-normal operators in semi-Hilbertian spaces. Aust. J. Math. Anal. Appl. 9(1), 5 (2012)

    MathSciNet  MATH  Google Scholar 

  29. Sain, D., Bhunia, P., Bhanja, A., Paul, K.: On a new norm on B(H) and its application to numerical radius inequalities. Ann. Funct. Anal. 12(4), 51 (2021). https://doi.org/10.1007/s43034-021-00138-5

    Article  MathSciNet  MATH  Google Scholar 

  30. Zamani, A., Moslehian, M.S., Xu, Q., Fu, C.: Numerical radius inequalities concerning with algebra norms. Mediterr. J. Math. 18(2), 38 (2021). https://doi.org/10.1007/s00009-020-01665-6

    Article  MathSciNet  MATH  Google Scholar 

  31. Zamani, A.: \(A\)-numerical radius inequalities for semi-Hilbertian space operators. Linear Algebra Appl. 578, 159–183 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Mr. Pintu Bhunia would like to thank UGC, Govt. of India for the financial support in the form of Senior Research Fellowship.

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Authors and Affiliations

  1. Department of Mathematics, Jadavpur University, Kolkata, 700032, India

    Pintu Bhunia & Kallol Paul

  2. Faculty of Economic Sciences and Management of Mahdia, University of Monastir, Mahdia, Tunisia

    Kais Feki

  3. Laboratory Physics-Mathematics and Applications (LR/13/ES-22), Faculty of Sciences of Sfax, University of Sfax, Sfax, Tunisia

    Kais Feki

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  1. Pintu Bhunia
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Correspondence to Kais Feki.

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Communicated by Mohammad Sal Moslehian.

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Bhunia, P., Feki, K. & Paul, K. Generalized A-Numerical Radius of Operators and Related Inequalities. Bull. Iran. Math. Soc. 48, 3883–3907 (2022). https://doi.org/10.1007/s41980-022-00727-7

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  • DOI: https://doi.org/10.1007/s41980-022-00727-7

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