Improvements of some Berezin radius inequalities

Year 2022, , 141 - 153, 15.09.2022

Mehmet Gürdal , Mohammad Alomarı

https://doi.org/10.33205/cma.1110550

Cited By: 3

Abstract

The Berezin transform $\widetilde{A}$ and the Berezin radius of an operator $A$ on the reproducing kernel Hilbert space over some set $Q$ with normalized reproducing kernel $k_{\eta}:=\dfrac{K_{\eta}}{\left\Vert K_{\eta}\right\Vert}$ are defined, respectively, by $\widetilde{A}(\eta)=\left\langle {A}k_{\eta},k_{\eta}\right\rangle$, $\eta\in Q$ and $\mathrm{ber} (A):=\sup_{\eta\in Q}\left\vert \widetilde{A}{(\eta)}\right\vert$. A simple comparison of these properties produces the inequalities $\dfrac{1}{4}\left\Vert A^{\ast}A+AA^{\ast}\right\Vert \leq\mathrm{ber}^{2}\left( A\right) \leq\dfrac{1}{2}\left\Vert A^{\ast}A+AA^{\ast}\right\Vert $. In this research, we investigate other inequalities that are related to them. In particular, for $A\in\mathcal{L}\left( \mathcal{H}\left(Q\right) \right) $ we prove that
$\mathrm{ber}^{2}\left( A\right) \leq\dfrac{1}{2}\left\Vert A^{\ast}A+AA^{\ast}\right\Vert _{\mathrm{ber}}-\dfrac{1}{4}\inf_{\eta\in Q}\left(\left( \widetilde{\left\vert A\right\vert }\left( \eta\right)\right)-\left( \widetilde{\left\vert A^{\ast}\right\vert }\left( \eta\right)\right) \right) ^{2}.$

Keywords

Mixed Schwarz inequality, Berezin radius, Furuta inequality

References

• M. W. Alomari: On the generalized mixed Schwarz inequality, Proc. Inst. Math. Mech., 46 (1) (2020), 3–15.
• M. W. Alomari: Refinements of some numerical radius inequalities for Hilbert space operators, Linear Multilinear Algebra, 69 (7) (2021), 1208–1223.
• M. W. Alomari: Improvements of some numerical radius inequalities, Azerb. J. Math., 12 (1) (2022), 124–137.
• M. Bakherad: Some Berezin number inequalities for operators matrices, Czechoslovak Math. J., 68 (143) (2018), 997–1009.
• M. Bakherad, M. T. Garayev: Berezin number inequalities for operators, Concr. Oper., 6 (1) (2019), 33–43.
• M. Bakherad, M. Hajmohamadi, R. Lashkaripour and S. Sahoo: Some extensions of Berezin number inequalities on operators, Rocky Mountain J. Math., 51 (6) (2021), 1941–1951.
• F. A. Berezin: Covariant and contravariant symbols for operators, Math. USSR-Izvestiya, 6 (1972), 1117–1151.
• S. S. Dragomir: Inequalities for the numerical radius of linear operators in Hilbert spaces, SpringerBriefs in Mathematics (2013).
• S. S. Dragomir: Some inequalities for the norm and the numerical radius of linear operators in Hilbert spaces, Tamkang J. Math., 39 (2008), 1–7.
• S. S. Dragomir: Some Inequalities generalizing Kato’s and Furuta’s results, Filomat, 28 (1) (2014), 179–195.
• T. Furuta: An extension of the Heinz-Kato theorem, Proc. Amer. Math. Soc., 120 (3) (1994), 785–787.
• M. T. Garayev, M. W. Alomari: Inequalities for the Berezin number of operators and related questions, Complex Anal. Oper. Theory, 15, 30 (2021).
• M. Garayev, F. Bouzeffour, M. Gürdal and C. M. Yangöz: Refinements of Kantorovich type, Schwarz and Berezin number inequalities, Extracta Math., 35 (2020), 1–20.
• M. T. Garayev, M. Gürdal and A. Okudan: Hardy-Hilbert’s inequality and a power inequality for Berezin numbers for operators, Math. Inequal. Appl., 19 (2016), 883–891.
• M. T. Garayev, M. Gürdal and S. Saltan: Hardy type inequaltiy for reproducing kernel Hilbert space operators and related problems, Positivity, 21 (6) (2017), 1615–1623.
• M. T. Garayev, H. Guedri, M. Gürdal and G. M. Alsahli: On some problems for operators on the reproducing kernel Hilbert space, Linear Multilinear Algebra, 69 (11) (2021), 2059–2077.
• K. E. Gustafson, D. K. M. Rao: Numerical Range, Springer-Verlag, New York (1997).
• M. Hajmohamadi, R. Lashkaripour and M. Bakherad: Improvements of Berezin number inequalities, Linear Multilinear Algebra, 68 (6) (2020), 1218–1229.
• P. R. Halmos: A Hilbert space problem book, Van Nostrand Company, Inc., Princeton (1967).
• M. B. Huban, H. Ba¸saran and M. Gürdal: New upper bounds related to the Berezin number inequalities, J. Inequal. Spec. Funct., 12 (3) (2021), 1–12.
• M. T. Karaev: Berezin set and Berezin number of operators and their applications, The 8th Workshop on Numerical Ranges and Numerical Radii WONRA -06, Bremen (Germany) (2006), p.14.
• M. T. Karaev: Berezin symbol and invertibility of operators on the functional Hilbert spaces, J. Funct. Anal., 238 (2006), 181–192.
• M. T. Karaev: Reproducing kernels and Berezin symbols techniques in various questions of operator theory, Complex Anal. Oper. Theory, 7 (2013), 983–1018.
• F. Kittaneh: A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix, Studia Math., 158 (2003), 11–17.
• F. Kittaneh: Numerical radius inequalities for Hilbert space operators, Studia Math., 168 (1) (2005), 73–80
• F. Kittaneh, Y. Manasrah: Improved Young and Heinz inequalities for matrices, J. Math. Anal. Appl., 361 (1) (2010), 262–269.
• T. Kato: Notes on some inequalities for linear operators, Math. Ann., 125 (1952), 208–212.
• W. Reid: Symmetrizable completely continuous linear transformations in Hilbert space, Duke Math., 18 (1951), 41–56.
• S. Sahoo, M. Bakherad: Some extended Berezin number inequalities, Filomat, 35 (6) (2021), 2043–2053.
• M. Sattari, M. S. Moslehian and T. Yamazaki: Some generalized numerical radius inequalities for Hilbert space operators, Linear Algebra Appl., 470 (2015), 216–227.
• A. Sheikhhosseini, M. S. Moslehian and K. Shebrawi: Inequalities for generalized Euclidean operator radius via Young’s inequality, J. Math. Anal. Appl., 445 (2) (2017), 1516–1529.
• R. Tapdigoglu: New Berezin symbol inequalities for operators on the reproducing kernel Hilbert space, Oper. Matrices, 15 (3) (2021), 1445–1460.
• U. Yamancı, M. Gürdal and M. T. Garayev: Berezin number inequality for convex function in reproducing kernel Hilbert space, Filomat, 31 (2017), 5711–5717.
• U. Yamancı, R. Tunç and M. Gürdal: Berezin numbers, Grüss type inequalities and their applications, Bull. Malays. Math. Sci. Soc., 43 (2020), 2287–2296.
• T. Yamazaki: On upper and lower bounds of the numerical radius and an equality condition, Studia Math., 178 (2007), 83–89.