More inequalities on numerical radii of sectorial matrices

Chaojun Yang; Chaojun Yang

doi:10.3934/math.2021233

AIMS Mathematics

2021, Volume 6, Issue 4: 3927-3939. doi: 10.3934/math.2021233

Previous Article Next Article

Research article

More inequalities on numerical radii of sectorial matrices

Chaojun Yang ^{1
,
,}

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, P. R. China

Received: 17 December 2020 Accepted: 28 January 2021 Published: 02 February 2021
MSC : 15A45, 15A60

In this article, we refine some numerical radius inequalities of sectorial matrices recently obtained by Bedrani, Kittaneh and Sababheh. Among other results, it is shown that if $ A_i\in\mathbb{M}_n(\mathbb{C}) $ with $ W(A_i)\subseteq S_{\alpha} $, $ i = 1, 2\cdots, n $, and $ a_1, \cdots, a_n $ are positive real numbers with $ \sum_{j = 1}^na_j = 1 $, then

$ \begin{eqnarray*} \omega^t\left(\sum\limits_{i = 1}^n a_iA_i\right)\le\cos^{2t}(\alpha)\omega\left(\sum\limits_{i = 1}^n a_iA_i^t\right), \end{eqnarray*} $

where $ t\in[-1, 0] $. An improvement of the Heinz-type inequality for the numerical radii of sectorial matrices is also given. Moreover, we present some numerical radius inequalities of sectorial matrices involving positive linear maps.
- sectorial matrices,
- numerical radius,
- function,
- operator mean,
- positive linear maps
Citation: Chaojun Yang. More inequalities on numerical radii of sectorial matrices[J]. AIMS Mathematics, 2021, 6(4): 3927-3939. doi: 10.3934/math.2021233

Related Papers:

Abstract

In this article, we refine some numerical radius inequalities of sectorial matrices recently obtained by Bedrani, Kittaneh and Sababheh. Among other results, it is shown that if $ A_i\in\mathbb{M}_n(\mathbb{C}) $ with $ W(A_i)\subseteq S_{\alpha} $, $ i = 1, 2\cdots, n $, and $ a_1, \cdots, a_n $ are positive real numbers with $ \sum_{j = 1}^na_j = 1 $, then

$ \begin{eqnarray*} \omega^t\left(\sum\limits_{i = 1}^n a_iA_i\right)\le\cos^{2t}(\alpha)\omega\left(\sum\limits_{i = 1}^n a_iA_i^t\right), \end{eqnarray*} $

where $ t\in[-1, 0] $. An improvement of the Heinz-type inequality for the numerical radii of sectorial matrices is also given. Moreover, we present some numerical radius inequalities of sectorial matrices involving positive linear maps.

References

[1]	T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra Appl., 26 (1979), 203–241. doi: 10.1016/0024-3795(79)90179-4
[2]	Y. Bedrani, F. Kittaneh, M. Sababheh, From positive to accretive matrices, 2020, ArXiv: 2002.11090.
[3]	Y. Bedrani, F. Kittaneh, M. Sababheh, Numerical radii of accretive matrices, Linear Multilinear A., 2020, DOI: 10.1080/03081087.2020.1813679.
[4]	Y. Bedrani, F. Kittaneh, M. Sababheh, On the weighted geometric mean of accretive matrices, Ann. Funct. Anal., 12 (2020), DOI: 10.1007/s43034-020-00094-6.
[5]	R. Bhatia, Positive definite matrices, Princeton: Princeton University Press, 2007.
[6]	R. Bhatia, Matrix analysis, New York: Springer-Verlag, 1997.
[7]	J. C. Bourin, E. Y. Lee, Unitary orbits of Hermitian operators with convex or concave functions, Bull. London Math. Soc., 44 (2012), 1085–1102. doi: 10.1112/blms/bds080
[8]	D. Choi, T. Y. Tam, P. Zhang, Extensions of Fischer's inequality, Linear Algebra Appl., 569 (2019), 311–322. doi: 10.1016/j.laa.2019.01.022
[9]	S. Drury, Principal powers of matrices with positive definite real part, Linear Multilinear Algebra, 63 (2015), 296–301. doi: 10.1080/03081087.2013.865732
[10]	S. Drury, M. H. Lin, Singular value inequalities for matrices with numerical ranges in a sector, Oper. Matrices, 8 (2014), 1143–1148.
[11]	M. Gaál, M. Pálfia, A note on real operator monotone functions, Int. Math. Res. Not., 2020, DOI: 10.1093/imrn/rnaa150.
[12]	R. A. Horn, C. R. Johnson, Matrix analysis, Cambridge: Cambridge University Press, 2013.
[13]	F. Kittaneh, M. Sakkijha, Inequalities for accretive-dissipative matrices, Linear Multilinear Algebra, 67 (2019), 1037–1042. doi: 10.1080/03081087.2018.1441800
[14]	M. H. Lin, Fischer type determinantal inequalities for accretive-dissipative matrices, Linear Algebra Appl., 438 (2013), 2808–2812. doi: 10.1016/j.laa.2012.11.016
[15]	M. H. Lin, Some inequalities for sector matrices, Oper. Matrices, 10 (2016), 915–921.
[16]	M. H. Lin, Extension of a result of Hanynsworth and Hartfiel, Arch. Math., 104 (2015), 93–100. doi: 10.1007/s00013-014-0717-2
[17]	M. H. Lin, D. M. Zhou, Norm inequalities for accretive-dissipative operator matrices, J. Math. Anal. Appl., 407 (2013), 436–442. doi: 10.1016/j.jmaa.2013.05.042
[18]	M. H. Lin, F. F. Sun, A property of the geometric mean of accretive operators, Linear Multilinear Algebra, 65 (2017), 433–437. doi: 10.1080/03081087.2016.1188878
[19]	Y. L. Mao, Y. P. Mao, Inequalities for the Heinz mean of sector matrices, Bull. Iran. Math. Soc., 46 (2020), 1767–1774. doi: 10.1007/s41980-020-00357-x
[20]	M. Raissouli, M. S. Moslehian, S. Furuichi, Relative entropy and Tsallis entropy of two accretive operators, C. R. Acad. Sci. Paris Ser. I, 355 (2017), 687–693. doi: 10.1016/j.crma.2017.05.005
[21]	F. Tan, H. Chen, Inequalities for sector matrices and positive linear maps, Electronic J. Linear Algebra, 35 (2019), 418–423.
[22]	F. Tan, A. Xie, An extension of the AM-GM-HM inequality, Bull. Iran. Math. Soc., 46 (2020), 245–251. doi: 10.1007/s41980-019-00253-z
[23]	C. Yang, F. Lu, Inequalities for the Heinz mean of sector matrices involving positive linear maps, Ann. Funct. Anal. 11 (2020), 866–878.
[24]	F. Zhang, A matrix decomposition and its applications, Linear Multilinear Algebra, 63 (2015), 2033–2042. doi: 10.1080/03081087.2014.933219
[25]	Y. Zheng, X. Jiang, X. Chen, A. Zhang, F. Alsaadi, Means and the Schur complement of sector matrices, Linear Multilinear Algebra, 2020, DOI: 10.1080/03081087.2020.1809617.

Reader Comments

Your name:*

Email:*
© 2021 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)