Some Refinements of the Numerical Radius Inequalities via Young Inequality


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Authors: Z. HEYDARBEYGI AND M. AMYARI

DOI: 10.46793/KgJMat2102.191H

Abstract:

In this paper, we get an improvement of the Hölder-McCarthy operator inequality in the case when r 1 and refine generalized inequalities involving powers of the numerical radius for sums and products of Hilbert space operators.



Keywords:

Bounded linear operator, Hilbert space, norm inequality, numerical radius inequality.



References:

[1]   S. S. Dragomir, Vector inequalities for powers of some operators in Hilbert spaces, Filomat 23(1) (2009), 69–83.

[2]   S. S. Dragomir, A note on new refinements and reverses of Young’s inequality, Transylvanian Journal of Mathematics and Mechanics 8(1)(2016), 46–49.

[3]   M. Fujii and R. Nakamoto, Refinements of Hölder-McCarthy inequality and Young inequality, Adv. Oper. Theory 1(2) (2016), 184–188.

[4]   K. E. Gustafson and D. K. M. Rao, Numerical Range, Springer-Verlag, New York, 1997.

[5]   J. A. R. Holbrook, Multiplicative properties of the numerical radius in operator theory, J. Reine Angew. Math. 237 (1969), 166–174.

[6]   M. Kian, Operator Jensen inequality for superquadratic functions, Linear Algebra Appl. 456 (2014), 82–87.

[7]   F. Kittaneh, A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix, Studia Math. 158(1) (2003), 11–17.

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

[9]   F. Kittaneh and M. El-Haddad, Numerical radius inequalities for Hilbert space operators II, Studia Math. 182(2) (2007), 133–140.

[10]   F. Kittaneh, M. S. Moslehian and T. Yamazaki, Cartesian decomposition and numerical radius inequalities, Linear Algebra Appl. 471 (2015), 46–53.

[11]   H. Kober, On the arithmetic and geometric means and on Hölder’s inequality, Proc. Amer. Math. Soc. 9 (1958), 452–459.

[12]   C-S. Lin and Y. J. Cho, On Hölder-McCarthy-type inequalities with powers, J. Korean Math. Soc. 39(3) (2002), 351–361.

[13]   M. Sattari, M. S. Moslehian and T. Yamazaki, Some generalized numerical radius inequalities for Hilbert space operators, Linear Algebra Appl. 470 (2015), 216–227.

[14]   M. Sababheh, Heinz-type numerical radii inequalities, Linear Multilinear Algebra, DOI 10.1080/03081087.2018.1440518.

[15]   M. Sababheh, Numerical radius inequalities via convexity, Linear Algebra Appl. 549 (2018), 67–78.

[16]   Y. Seo, Hölder type inequalities on Hilbert C-modules and its reverses, Ann. Funct. Anal. 5(1) (2014), 1–9.

[17]   K. Shebrawi and H. Albadwi, Numerical radius and operator norm inequalities, J. Inequal. Appl. (2009), Article ID 492154, 11 pages.

[18]   K. Shebrawi, Numerical radius inequalities for certain 2 × 2 operator matrices II, Linear Algebra Appl. 523 (2017), 1–12.

[19]   A. Zamani, Some lower bounds for the numerical radius of Hilbert space operators, Adv. Oper. Theory 2(2) (2007), 98–107.