Novel Berezin number and norm inequalities for operator sums and products

Feryal Aladsani; Asmahan Alajyan; Salma Aljawi; Kais Feki; Feryal Aladsani; Asmahan Alajyan; Salma Aljawi; Kais Feki

doi:10.3934/math.2026236

AIMS Mathematics

2026, Volume 11, Issue 3: 5738-5758. doi: 10.3934/math.2026236

Previous Article Next Article

Research article Special Issues

Novel Berezin number and norm inequalities for operator sums and products

1.
Department of Mathematics and Statistics, College of Science, King Faisal University, Hafuf 31982, Al Ahsa, Saudi Arabia
2.
Department of Mathematical Sciences, College of Science, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
3.
Department of Mathematics, College of Science and Arts, Najran University, Najran 66462, Saudi Arabia
4.
Science and Engineering Research Center, Najran University, Najran, Saudi Arabia

Received: 11 November 2025 Revised: 06 February 2026 Accepted: 14 February 2026 Published: 06 March 2026
MSC : 26D15, 46C05, 47A12, 47A30, 47A63

Let $ (\mathcal{X}_\mathscr{F}, \langle \cdot, \cdot \rangle) $ be a reproducing kernel Hilbert space over a non-empty set $ \mathscr{F} $. Let $ \widehat{u}_\lambda $ and $ \widehat{u}_\mu $ denote the normalized reproducing kernels of $ \mathcal{X}_\mathscr{F} $. The Berezin number and the Berezin norm of a bounded linear operator $ \mathcal{B} $ acting on $ \mathcal{X}_\mathscr{F} $ are, respectively, defined by
$ \mathbf{ber}(\mathcal{B}) = \sup\limits_{\lambda \in \mathscr{F}} \big|\langle \mathcal{B} \widehat{u}_\lambda, \widehat{u}_\lambda \rangle\big| \quad \text{and} \quad \|\mathcal{B}\|_{\mathbf{ber}} = \sup\limits_{\lambda, \mu \in \mathscr{F}} \big|\langle \mathcal{B} \widehat{u}_\lambda, \widehat{u}_\mu \rangle\big|. $
In this work, we establish new upper bounds for these two quantities. In particular, we derive bounds for their sums and obtain novel estimates for a specific type of product, namely $ \mathbf{ber}(\mathcal{C}^*\mathcal{B}) $, where $ \mathcal{C}^* $ denotes the adjoint of $ \mathcal{C} $. Some of our results also involve another Berezin-type norm that is equivalent to the quantities mentioned above. Several applications and improvements of existing results in the literature are provided.
- Berezin number,
- Berezin norm,
- reproducing kernel Hilbert space,
- operator inequalities,
- numerical radius,
- bounded linear operators
Citation: Feryal Aladsani, Asmahan Alajyan, Salma Aljawi, Kais Feki. Novel Berezin number and norm inequalities for operator sums and products[J]. AIMS Mathematics, 2026, 11(3): 5738-5758. doi: 10.3934/math.2026236

Related Papers:

Abstract

Let $ (\mathcal{X}_\mathscr{F}, \langle \cdot, \cdot \rangle) $ be a reproducing kernel Hilbert space over a non-empty set $ \mathscr{F} $. Let $ \widehat{u}_\lambda $ and $ \widehat{u}_\mu $ denote the normalized reproducing kernels of $ \mathcal{X}_\mathscr{F} $. The Berezin number and the Berezin norm of a bounded linear operator $ \mathcal{B} $ acting on $ \mathcal{X}_\mathscr{F} $ are, respectively, defined by

$ \mathbf{ber}(\mathcal{B}) = \sup\limits_{\lambda \in \mathscr{F}} \big|\langle \mathcal{B} \widehat{u}_\lambda, \widehat{u}_\lambda \rangle\big| \quad \text{and} \quad \|\mathcal{B}\|_{\mathbf{ber}} = \sup\limits_{\lambda, \mu \in \mathscr{F}} \big|\langle \mathcal{B} \widehat{u}_\lambda, \widehat{u}_\mu \rangle\big|. $

In this work, we establish new upper bounds for these two quantities. In particular, we derive bounds for their sums and obtain novel estimates for a specific type of product, namely $ \mathbf{ber}(\mathcal{C}^*\mathcal{B}) $, where $ \mathcal{C}^* $ denotes the adjoint of $ \mathcal{C} $. Some of our results also involve another Berezin-type norm that is equivalent to the quantities mentioned above. Several applications and improvements of existing results in the literature are provided.

References

[1]	P. R. Halmos, A Hilbert space problem book, 2 Eds., New York: Springer, 1982.
[2]	K. E. Gustafson, D. K. M. Rao, Numerical range, New York: Springer, 1997. https://doi.org/10.1007/978-1-4613-8498-4
[3]	W. Bani-Domi, F. Kittaneh, Norm and numerical radius inequalities for Hilbert space operators, Linear Multilinear Algebra, 69 (2021), 934–945. https://doi.org/10.1080/03081087.2020.1798334 doi: 10.1080/03081087.2020.1798334
[4]	P. Bhunia, K. Paul, New upper bounds for the numerical radius of Hilbert space operators, Bull. Sci. Math., 167 (2021), 102959. https://doi.org/10.1016/j.bulsci.2021.102959 doi: 10.1016/j.bulsci.2021.102959
[5]	N. Altwaijry, K. Feki, N. Minculete, Further inequalities for the weighted numerical radius of operators, Mathematics, 10 (2022), 1–17. https://doi.org/10.3390/math10193576 doi: 10.3390/math10193576
[6]	M. L. Arias, G. Corach, M. C. Gonzalez, Partial isometries in semi-Hilbertian spaces, Linear Algebra Appl., 428 (2008), 1460–1475. https://doi.org/10.1016/j.laa.2007.09.031 doi: 10.1016/j.laa.2007.09.031
[7]	M. L. Arias, G. Corach, M. C. Gonzalez, Metric properties of projections in semi-Hilbertian spaces, Integral Equ. Oper. Theory, 62 (2008), 11–28.
[8]	P. Bhunia, F. Kittaneh, K. Paul, A. Sen, Anderson's theorem and $A$-spectral radius bounds for semi-Hilbertian space operators, Linear Algebra Appl., 657 (2023), 147–162. https://doi.org/10.1016/j.laa.2022.10.019 doi: 10.1016/j.laa.2022.10.019
[9]	M. Guesba, M. Sababheh, On $A$-numerical radius inequalities of semi-Hilbert space operators, Ann. Univ. Ferrara, 71 (2025), 58. https://doi.org/10.1007/s11565-025-00613-0 doi: 10.1007/s11565-025-00613-0
[10]	A. Zamani, $A$-numerical radius inequalities for semi-Hilbertian space operators, Linear Algebra Appl., 578 (2019), 159–183. https://doi.org/10.1016/j.laa.2019.05.012 doi: 10.1016/j.laa.2019.05.012
[11]	A. Saddi, $A$-normal operators in semi-Hilbertian spaces, Aust. J. Math. Anal. Appl., 9 (2012), 1–12.
[12]	N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), 337–404. https://doi.org/10.1090/S0002-9947-1950-0051437-7 doi: 10.1090/S0002-9947-1950-0051437-7
[13]	K. H. Zhu, Operator theory in function spaces, 2 Eds., American Mathematical Society, 2007.
[14]	V. I. Paulsen, M. Raghupathi, An introduction to the theory of reproducing kernel Hilbert spaces, Cambridge University Press, 2016.
[15]	F. A. Berezin, Covariant and contravariant symbols for operators, Math. USSR Izv., 6 (1972), 1117–1151. https://doi.org/10.1070/IM1972v006n05ABEH001913 doi: 10.1070/IM1972v006n05ABEH001913
[16]	F. A. Berezin, Quantization, Math. USSR Izv., 8 (1974), 1109–1165. https://doi.org/10.1070/IM1974v008n05ABEH002140 doi: 10.1070/IM1974v008n05ABEH002140
[17]	M. T. Karaev, Reproducing kernels and Berezin symbols techniques in various questions of operator theory, Complex Anal. Oper. Theory, 7 (2013), 983–1018. https://doi.org/10.1007/s11785-012-0232-z doi: 10.1007/s11785-012-0232-z
[18]	M. B. Huban, H. Başaran, M. Gürdal, New upper bounds related to the Berezin number inequalities, J. Inequal. Spec. Funct., 12 (2021), 1–12.
[19]	M. T. Garayev, M. W. Alomari, Inequalities for the Berezin number of operators and related questions, Complex Anal. Oper. Theory, 15 (2021), 30. https://doi.org/10.1007/s11785-021-01078-7 doi: 10.1007/s11785-021-01078-7
[20]	S. Majee, A. Maji, A. Manna, Numerical radius and Berezin number inequality, J. Math. Anal. Appl., 517 (2023), 126566. https://doi.org/10.1016/j.jmaa.2022.126566 doi: 10.1016/j.jmaa.2022.126566
[21]	M. Bakherad, M. T. Karaev, Berezin number inequalities for operators, Concr. Oper., 6 (2019), 33–43. https://doi.org/10.1515/conop-2019-0003 doi: 10.1515/conop-2019-0003
[22]	M. Garayev, S. Saltan, F. Bouzeffour, B. Aktan, Some inequalities involving Berezin symbols of operator means and related questions, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 114 (2020), 85. https://doi.org/10.1007/s13398-020-00815-5 doi: 10.1007/s13398-020-00815-5
[23]	P. Bhunia, K. Paul, A. Sen, Inequalities involving Berezin norm and Berezin number, Complex Anal. Oper. Theory, 17 (2023), 7. https://doi.org/10.1007/s11785-022-01305-9 doi: 10.1007/s11785-022-01305-9
[24]	F. Chien, M. Bakherad, M. W. Alomari, Refined Berezin number inequalities via superquadratic and convex functions, Filomat, 37 (2023), 265–277. https://doi.org/10.2298/FIL2301265C doi: 10.2298/FIL2301265C
[25]	P. Bhunia, A. Sen, K. Paul, Development of the Berezin number inequalities, Acta Math. Sin. Engl. Ser., 39 (2023), 1219–1228. https://doi.org/10.1007/s10114-023-2090-1 doi: 10.1007/s10114-023-2090-1
[26]	M. T. Karaev, Berezin symbol and invertibility of operators on the functional Hilbert spaces, J. Funct. Anal., 238 (2006), 181–192. https://doi.org/10.1016/j.jfa.2006.04.030 doi: 10.1016/j.jfa.2006.04.030
[27]	M. T. Karaev, S. Saltan, Some results on Berezin symbols, Complex Var. Theory Appl., 50 (2005), 185–193. https://doi.org/10.1080/02781070500032861 doi: 10.1080/02781070500032861
[28]	E. Nordgren, P. Rosenthal, Boundary values of Berezin symbols, In: Nonselfadjoint operators and related topics, 73 (1994), 362–368. https://doi.org/10.1007/978-3-0348-8522-5_14
[29]	C. Conde, K. Feki, F. Kittaneh, Berezin number and norm inequalities for operators in Hilbert and semi-Hilbert spaces, In: Matrix and operator equations and applications, Cham: Springer, 2023,525–558. https://doi.org/10.1007/16618_2023_55
[30]	M. W. Alomari, On Cauchy-Schwarz type inequalities and applications to numerical radius inequalities, Ricerche Mat., 73 (2024), 1493–1510. https://doi.org/10.1007/s11587-022-00689-2 doi: 10.1007/s11587-022-00689-2
[31]	N. Altwaijry, K. Feki, N. Minculete, Numerical radius, Berezin number, and Berezin norm inequalities for sums of operators, Turk. J. Math., 47 (2023), 1481–1497. https://doi.org/10.55730/1300-0098.3442 doi: 10.55730/1300-0098.3442
[32]	N. Altwaijry, K. Feki, N. Minculete, On some generalizations of Cauchy-Schwarz inequalities and their applications, Symmetry, 15 (2023), 304. https://doi.org/10.3390/sym15020304 doi: 10.3390/sym15020304
[33]	J. Pečarić, T. Furuta, J. M. Hot, Y. Seo, Mond-Pečarić method in operator inequalities, 2 Eds., Zagreb: Element, 2005.
[34]	M. L. Buzano, Generalizzazione della diseguaglianza di Cauchy-Schwarz, Rend. Sem. Mat. Univ. Politech. Torino, 31 (1974), 405–409.

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)