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Journals / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 53 Issue: 1

Inequalities for the $A$-joint numerical radius of two operators and their applications

Kais Feki(Najran University)0000-0002-9326-4173
Pages
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Publication Date
2024-02-29
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AI Summary & Key Takeaways

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⚡ TL;DR(Too Long; Didn't Read)

Moreover, several bounds related to the $A$-Davis-Wielandt radius of semi-Hilbert space operators are also provided.

📌 3 Key Takeaways

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Objective & Hypothesis:Let $\big(\mathcal{H}, \langle \cdot, \cdot\rangle \big)$ be a complex Hilbert space and $A$ be a positive (semidefinite) bounded linear operator on $\mathcal{H}$.
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Key Findings & Data:The $A$-joint numerical radius of two $A$-bounded operators $T$ and $S$ is given by \begin{align*} \omega_{A,\text{e}}(T,S) = \sup_{\|x\|_A= 1}\sqrt{\big|{\langle Tx, x\rangle}_A\big|^2+\big|{\langle Sx, x\rangle}_A\big|^2}. \end{align*} In this paper, we aim to prove several bounds involving $\omega_{A,\text{e}}(T,S)$.
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Conclusion & Impact:Moreover, several bounds related to the $A$-Davis-Wielandt radius of semi-Hilbert space operators are also provided.
🔬Design:Research Article
📍Scope:2005 Verileri
⚙️Variables:Semi-inner product · positive operator · $A$-joint numerical radius

Abstract

Let $\big(\mathcal{H}, \langle \cdot, \cdot\rangle \big)$ be a complex Hilbert space and $A$ be a positive (semidefinite) bounded linear operator on $\mathcal{H}$. The semi-inner product induced by $A$ is given by ${\langle x, y\rangle}_A := \langle Ax, y\rangle$, $x, y\in\mathcal{H}$ and defines a seminorm ${\|\cdot\|}_A$ on $\mathcal{H}$. This makes $\mathcal{H}$ into a semi-Hilbert space. The $A$-joint numerical radius of two $A$-bounded operators $T$ and $S$ is given by \begin{align*} \omega_{A,\text{e}}(T,S) = \sup_{\|x\|_A= 1}\sqrt{\big|{\langle Tx, x\rangle}_A\big|^2+\big|{\langle Sx, x\rangle}_A\big|^2}. \end{align*} In this paper, we aim to prove several bounds involving $\omega_{A,\text{e}}(T,S)$. This allows us to establish some inequalities for the $A$-numerical radius of $A$-bounded operators. In particular, we extend the well-known inequalities due to Kittaneh [Numerical radius inequalities for Hilbert space operators, Studia Math. 168 (1), 73-80, 2005]. Moreover, several bounds related to the $A$-Davis-Wielandt radius of semi-Hilbert space operators are also provided.

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Cite This Article

Feki, K. (2024). Inequalities for the $A$-joint numerical radius of two operators and their applications. Hacettepe Journal of Mathematics and Statistics, Volume: 53 Issue: 1. https://doi.org/10.15672/hujms.1142554

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References (26)

  1. N. Altwaijry, K. Feki and N. Minculete, Further inequalities for the weighted numerical radius of operators, Mathematics, 10 (19), 3576, 2022.
  2. N. Altwaijry, K. Feki and N. Minculete, On Some Generalizations of Cauchy–Schwarz Inequalities and Their Applications, Symmetry, 15(2), 304, 2023.
  3. M.L. Arias, G. Corach and M.C. Gonzalez, Partial isometries in semi-Hilbertian spaces, Linear Algebra Appl. 428 (7), 1460–1475, 2008.
  4. M.L. Arias, G. Corach and M.C. Gonzalez, Metric properties of projections in semi-Hilbertian spaces, Integral Equations and Operator Theory, 62, 11–28, 2008.
  5. H. Baklouti, K. Feki and O.A.M. Sid Ahmed, Joint numerical ranges of operators in semi-Hilbertian spaces, Linear Algebra Appl. 555, 266–284, 2018.
  6. H. Baklouti and S. Namouri, Spectral analysis of bounded operators on semi-Hilbertian spaces, Banach J. Math. Anal. 16, 12, 2022.
  7. P. Bhunia, S. S. Dragomir, M. S. Moslehian and K. Paul, Lectures on numerical radius inequalities, Infosys Science Foundation Series in Mathematical Sciences. Springer, 2022.
  8. P. Bhunia, K. Feki and K. Paul, Generalized A-numerical radius of operators and related inequalities, Bull. Iran. Math. Soc. 48, 3883–3907, 2022.
  9. P. Bhunia, F. Kittaneh, K. Paul and A. Sen, Anderson’s theorem and A-spectral radius bounds for semi-Hilbertian space operators, Linear Algebra Appl. 657, 147–162, 2023.
  10. M. L. Buzano, Generalizzazione della diseguaglianza di Cauchy-Schwarz (Italian), Rend. Sem. Mat. Univ. e Politech. Torino 31, 405-409, 1974.
  11. C. Conde and K. Feki, On some inequalities for the generalized joint numerical radius of semi-Hilbert space operators, Ricerche mat. 2021, doi:10.1007/s11587-021-00629-6.
  12. R.G. Douglas, On majorization, factorization and range inclusion of operators in Hilbert space, Proc. Amer. Math. Soc. 17, 413–416, 1966.
  13. S. S. Dragomir, Some inequalities for the Euclidean operator radius of two operators in Hilbert spaces, Linear Algebra Appl. 419, 256–264, 2006.
  14. S. S. Dragomir, Advances in inequalities of the Schwarz, triangle and Heisenberg type in inner product spaces, Nova Science Publishers, Inc., New York, 2007.
  15. K. Feki, Spectral radius of semi-Hilbertian space operators and its applications, Ann. Funct. Anal. 11, 929–946, 2020.
  16. K. Feki, A note on the A-numerical radius of operators in semi-Hilbert spaces, Arch. Math. 115, 535–544, 2020.
  17. K. Feki, On tuples of commuting operators in positive semidefinite inner product spaces, Linear Algebra Appl. 603, 313–328, 2020.
  18. K. Feki, Some numerical radius inequalities for semi-Hilbert space operators, J. Korean Math. Soc., 58 (6), 1385–1405, 2021.
  19. K. Feki, Further improvements of generalized numerical radius inequalities for semi-Hilbertian space operators, Miskolc Mathematical Notes 23 (2), 651–665, 2022.
  20. K. Feki and S. Sahoo, Further inequalities for the A-numerical radius of certain $2\times 2$ operator matrices, Georgian Mathematical Journal, 2022, doi:10.1515/gmj-2022-2204.
  21. K. Feki and O.A.M. Sid Ahmed, Davis-Wielandt shells of semi-Hilbertian space operators and its applications, Banach J. Math. Anal. 14, 1281–1304, 2020.
  22. F. Kittaneh, Numerical radius inequalities for Hilbert space operators, Studia Math. 168 (1), 73-80, 2005.
  23. F. Kittaneh and A. Zamani, Bounds for A-numerical radius based on an extension of A-Buzano inequality, J. Comput. Appl. Math. 2023, doi:10.1016/j.cam.2023.115070.
  24. A. Saddi, A-Normal operators in Semi-Hilbertian spaces, Aust. J. Math. Anal. Appl. 9 (1), 1–12, 2012.
  25. A. Zamani, A-numerical radius inequalities for semi-Hilbertian space operators, Linear Algebra Appl. 578, 159–183, 2019.
  26. A. Zamani and K. Shebrawi, Some upper bounds for the Davis-Wielandt radius of Hilbert space operators, Mediterr. J. Math. 17, 25, 2020.