Spectral estimates for multiparametric operator products via the $ \mathbb{A} $-Berezin norm in RKHS

Xiu-Liang Qiu; Mehmet Gürdal; Selim Çetin; Ömer Kişi; Qing-Bo Cai; Xiu-Liang Qiu; Mehmet Gürdal; Selim Çetin; Ömer Kişi; Qing-Bo Cai

doi:10.3934/math.2026256

AIMS Mathematics

2026, Volume 11, Issue 3: 6217-6230. doi: 10.3934/math.2026256

Previous Article Next Article

Research article Special Issues

Spectral estimates for multiparametric operator products via the $ \mathbb{A} $-Berezin norm in RKHS

1.
Chengyi College, Jimei University, Xiamen 361021, China
2.
Department of Mathematics, Suleyman Demirel University, 32260, Isparta, Turkey
3.
Department of Mathematics, Burdur Mehmet Akif Ersoy University, Burdur, Turkey
4.
Department of Mathematics, Bartın University, Bartın, Turkey
5.
School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, China

Received: 07 January 2026 Revised: 09 February 2026 Accepted: 02 March 2026 Published: 11 March 2026
MSC : 46E22, 47A12, 47B32

This paper addresses the spectral analysis of operators acting on reproducing kernel Hilbert spaces equipped with a semi-inner product induced by a positive operator $ \mathbb{A} $. A fundamental challenge in this setting is the geometric discrepancy between the normalized reproducing kernels and the unit $ \mathbb{A} $-sphere, which renders classical numerical radius techniques inapplicable. By overcoming this structural obstacle, we establish sharp inequalities for the $ \mathbb{A} $-Berezin number and $ \mathbb{A} $-Berezin norm. Our main contribution involves the derivation of multiparametric estimates for triple operator products of the form $ \mathrm{P}^{\alpha}X\mathrm{R}^{\alpha} $ involving Schatten-type exponents. These results generalize and refine existing bounds in the literature. Furthermore, we provide a qualitative analysis of the obtained bounds through weighted Toeplitz operators on Hardy spaces and verify the theoretical findings with concrete matrix examples involving the geometric behavior of weight functions.
- Berezin number,
- reproducing kernel Hilbert space,
- semi-inner product,
- $ \mathbb{A} $-Berezin number,
- multiparametric inequalities,
- Schatten-type exponents
Citation: Xiu-Liang Qiu, Mehmet Gürdal, Selim Çetin, Ömer Kişi, Qing-Bo Cai. Spectral estimates for multiparametric operator products via the $ \mathbb{A} $-Berezin norm in RKHS[J]. AIMS Mathematics, 2026, 11(3): 6217-6230. doi: 10.3934/math.2026256

Related Papers:

Abstract

This paper addresses the spectral analysis of operators acting on reproducing kernel Hilbert spaces equipped with a semi-inner product induced by a positive operator $ \mathbb{A} $. A fundamental challenge in this setting is the geometric discrepancy between the normalized reproducing kernels and the unit $ \mathbb{A} $-sphere, which renders classical numerical radius techniques inapplicable. By overcoming this structural obstacle, we establish sharp inequalities for the $ \mathbb{A} $-Berezin number and $ \mathbb{A} $-Berezin norm. Our main contribution involves the derivation of multiparametric estimates for triple operator products of the form $ \mathrm{P}^{\alpha}X\mathrm{R}^{\alpha} $ involving Schatten-type exponents. These results generalize and refine existing bounds in the literature. Furthermore, we provide a qualitative analysis of the obtained bounds through weighted Toeplitz operators on Hardy spaces and verify the theoretical findings with concrete matrix examples involving the geometric behavior of weight functions.

References

[1]	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
[2]	M. Gürdal, Description of extended eigenvalues and extended eigenvectors of integration operators on the Wiener algebra, Expo. Math., 27 (2009), 153–160. https://doi.org/10.1016/j.exmath.2008.10.006 doi: 10.1016/j.exmath.2008.10.006
[3]	M. Gürdal, M. T. Garayev, S. Saltan, Some concrete operators and their properties, Turkish J. Math., 39 (2015), 970–989. https://doi.org/10.3906/mat-1502-48 doi: 10.3906/mat-1502-48
[4]	M. T. Garayev, H. Guediri, M. Gürdal, G. M. Alsahli, On some problems for operators on the reproducing kernel Hilbert space, Linear Multilinear Algebra, 69 (2021), 2059–2077. https://doi.org/10.1080/03081087.2019.1659220 doi: 10.1080/03081087.2019.1659220
[5]	M. Karaev, M. Gürdal, S. Saltan, Some applications of Banach algebra techniques, Math. Nachr., 284 (2011), 1678–1689. https://doi.org/10.1002/mana.200910129 doi: 10.1002/mana.200910129
[6]	M. Bakherad, Some Berezin number inequalities for operator matrices, Czechoslovak Math. J., 68 (2018), 997–1009. https://doi.org/10.21136/CMJ.2018.0048-17 doi: 10.21136/CMJ.2018.0048-17
[7]	H. Başaran, M. Gürdal, A. N. Güncan, Some operator inequalities associated with Kantorovich and Hölder-McCarthy inequalities and their applications, Turkish J. Math., 43 (2019), 523–532. https://doi.org/10.3906/mat-1811-10 doi: 10.3906/mat-1811-10
[8]	P. Bhunia, M. Gürdal, K. Paul, A. Sen, R. Tapdigoglu, On a new norm on the space of reproducing kernel Hilbert space operators and Berezin radius inequalities, Numer. Funct. Anal. Optim., 44 (2023), 970–986. https://doi.org/10.1080/01630563.2023.2221857 doi: 10.1080/01630563.2023.2221857
[9]	U. Yamancı, M. Gürdal, On numerical radius and Berezin number inequalities for reproducing kernel Hilbert space, New York J. Math., 23 (2017), 1531–1537.
[10]	U. Yamancı, R. Tunç, M. Gürdal, Berezin number, Grüss-type inequalities and their applications, Bull. Malays. Math. Sci. Soc., 43 (2020), 2287–2296. https://doi.org/10.1007/s40840-019-00804-x doi: 10.1007/s40840-019-00804-x
[11]	R. Tapdigoglu, M. Gürdal, N. Altwaijry, N. Sari, Davis-Wielandt-Berezin radius inequalities via Dragomir inequalities, Oper. Matrices, 15 (2021), 1445–1460. http://dx.doi.org/10.7153/oam-2021-15-90 doi: 10.7153/oam-2021-15-90
[12]	R. Bilal, J. Hyder, M. S. Akram, S. S. Dragomir, On some new upper bounds for numerical radius, Asian Eur. J. Math., 18 (2025), 2450142. https://doi.org/10.1142/S1793557124501420 doi: 10.1142/S1793557124501420
[13]	V. Stojiljković, S. S. Dragomir, Refinement of the Cauchy-Schwarz inequality with refinements and generalizations of the numerical radius type inequalities for operators, Ann. Math. Comput. Sci., 21 (2024), 33–43. https://doi.org/10.56947/amcs.v21.246 doi: 10.56947/amcs.v21.246
[14]	A. Gourty, M. A. Ighachane, F. Kittaneh, Further $A$-numerical radius inequalities for semi-Hilbertian space operators, J. Anal., 33 (2025), 2023–2045. https://doi.org/10.1007/s41478-025-00901-0 doi: 10.1007/s41478-025-00901-0
[15]	F. Kittaneh, A. Zamani, A refinement of $A$-Buzano inequality and applications to $A$-numerical radius inequalities, Linear Algebra Appl., 697 (2024), 32–48. https://doi.org/10.1016/j.laa.2023.02.020 doi: 10.1016/j.laa.2023.02.020
[16]	S. S. Dragomir, Semi-inner products and applications, 2003.
[17]	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
[18]	K. Feki, Spectral radius of semi-Hilbertian space operators and its applications, Ann. Funct. Anal., 11 (2020), 929–946. https://doi.org/10.1007/s43034-020-00064-y doi: 10.1007/s43034-020-00064-y
[19]	K. Feki, A note on the $A$-numerical radius of operators in semi-Hilbert spaces, Arch. Math., 115 (2020), 535–544. https://doi.org/10.1007/s00013-020-01482-z doi: 10.1007/s00013-020-01482-z
[20]	F. Kittaneh, A. Zamani, Bounds for $\mathbb{A}$-numerical radius based on an extension of A-Buzano inequality, J. Comput. Appl. Math., 426 (2023), 115070. https://doi.org/10.1016/j.cam.2023.115070 doi: 10.1016/j.cam.2023.115070
[21]	M. H. M. Rashid, Inequalities for the $A$-norm and $A$-numerical radius of operator sums in semi-Hilbertian spaces with applications, Rend. Circ. Mat. Palermo Ser. 2, 75 (2026), 40. https://doi.org/10.1007/s12215-025-01353-y doi: 10.1007/s12215-025-01353-y
[22]	M. Gürdal, H. Başaran, $A$-Berezin number of operators, Proc. Inst. Math. Mech., 48 (2022), 77–87. https://doi.org/10.30546/2409-4994.48.1.2022.77 doi: 10.30546/2409-4994.48.1.2022.77
[23]	M. Gürdal, H. Başaran, On $A$-Berezin number in functional Hilbert space, Filomat, 38 (2024), 7657–7671. https://doi.org/10.2298/FIL2421657G doi: 10.2298/FIL2421657G
[24]	M. Gürdal, H. Başaran, On inequalities for $A$-Berezin radius of operators, Afrika Mat., 35 (2024), 44. https://doi.org/10.1007/s13370-024-01186-5 doi: 10.1007/s13370-024-01186-5
[25]	M. T. Karaev, M. Gürdal, M. B. Huban, Reproducing kernels, Engliš algebras and some applications, Stud. Math., 232 (2016), 113–141.
[26]	M. L. Arias, G. Corach, M. C. Gonzales, 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
[27]	R. Douglas, On majoration, factorization, and range inclusion of operators in Hilbert space, Proc. Amer. Math. Soc., 17 (1966), 413–415. https://doi.org/10.1090/S0002-9939-1966-0203464-1 doi: 10.1090/S0002-9939-1966-0203464-1
[28]	H. W. Qiao, G. J. Hai, A. Chen, Improvements of $A$-numerical radius for semi-Hilbertian space operators, J. Math. Inequal., 18 (2024), 791–810. http://dx.doi.org/10.7153/jmi-2024-18-43 doi: 10.7153/jmi-2024-18-43
[29]	Y. Al-Manasrah, F. Kittaneh, A generalization of two refined Young inequalities, Positivity, 19 (2015), 757–768. https://doi.org/10.1007/s11117-015-0326-8 doi: 10.1007/s11117-015-0326-8

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)