Some operator mean inequalities for sector matrices

Chaojun Yang; Chaojun Yang

doi:10.3934/math.2022602

AIMS Mathematics

2022, Volume 7, Issue 6: 10778-10789. doi: 10.3934/math.2022602

Previous Article Next Article

Research article

Some operator mean inequalities for sector matrices

Chaojun Yang ^,

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

Received: 26 January 2022 Revised: 04 March 2022 Accepted: 16 March 2022 Published: 31 March 2022
MSC : 15A45, 47A63

In this article, we obtain some operator mean inequalities of sectorial matrices involving operator monotone functions. Among other results, it is shown that if $ A, B\in\mathbb{M}_n(\mathbb{C}) $ are such that $ W(A), W(B)\subseteq S_{\alpha} $, $ f, g, h\in\mathfrak{m} $ are such that $ g^{\prime}(1) = h^{\prime}(1) = t $ for some $ t\in(0, 1) $ and $ 0 < mI_n\le \Re A, \Re B\le MI_n $, then

$ \begin{eqnarray*} \Re(\Phi(f(A))\sigma_h\Phi(f(B)))\le\sec^4(\alpha)K\Re \Phi(f(A\sigma_gB)), \end{eqnarray*} $

where $ M, m $ are scalars and $ \mathfrak{m} $ is the collection of all operator monotone function $ \varphi:(0, \infty)\rightarrow (0, \infty) $ satisfying $ \varphi(1) = 1 $. Moreover, we refine a norm inequality of sectorial matrices involving positive linear maps, which is a result of Bedrani, Kittaneh and Sababheh.
- sector matrices,
- operator monotone function,
- operator mean,
- numerical radius,
- positive linear maps
Citation: Chaojun Yang. Some operator mean inequalities for sector matrices[J]. AIMS Mathematics, 2022, 7(6): 10778-10789. doi: 10.3934/math.2022602

Related Papers:

Abstract

In this article, we obtain some operator mean inequalities of sectorial matrices involving operator monotone functions. Among other results, it is shown that if $ A, B\in\mathbb{M}_n(\mathbb{C}) $ are such that $ W(A), W(B)\subseteq S_{\alpha} $, $ f, g, h\in\mathfrak{m} $ are such that $ g^{\prime}(1) = h^{\prime}(1) = t $ for some $ t\in(0, 1) $ and $ 0 < mI_n\le \Re A, \Re B\le MI_n $, then

$ \begin{eqnarray*} \Re(\Phi(f(A))\sigma_h\Phi(f(B)))\le\sec^4(\alpha)K\Re \Phi(f(A\sigma_gB)), \end{eqnarray*} $

where $ M, m $ are scalars and $ \mathfrak{m} $ is the collection of all operator monotone function $ \varphi:(0, \infty)\rightarrow (0, \infty) $ satisfying $ \varphi(1) = 1 $. Moreover, we refine a norm inequality of sectorial matrices involving positive linear maps, which is a result of Bedrani, Kittaneh and Sababheh.

References

[1]	T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra Appl., 26 (1979), 203–241. https://doi.org/10.1016/0024-3795(79)90179-4 doi: 10.1016/0024-3795(79)90179-4
[2]	T. Ando, F. Hiai, Operator log-convex functions and operator means, Math. Ann., 350 (2011), 611–630. https://doi.org/10.1007/s00208-010-0577-4 doi: 10.1007/s00208-010-0577-4
[3]	Y. Bedrani, F. Kittaneh, M. Sababheh, From positive to accretive matrices, Positivity, 25 (2021), 1601–1629. https://doi.org/10.1007/s11117-021-00831-8 doi: 10.1007/s11117-021-00831-8
[4]	Y. Bedrani, F. Kittaneh, M. Sababheh, Numerical radii of accretive matrices, Linear Multilinear A., 69 (2021), 957–970. https://doi.org/10.1080/03081087.2020.1813679 doi: 10.1080/03081087.2020.1813679
[5]	R. Bhatia, Positive definite matrices, Princeton: Princeton University Press, 2007. https://doi.org/10.1515/9781400827787
[6]	R. Bhatia, Matrix analysis, New York: Springer-Verlag, 1997. https://doi.org/10.1007/978-1-4612-0653-87
[7]	R. Bhatia, F. Kittaneh, Notes on matrix arithmetic-geometric mean inequalities, Linear Algebra Appl., 308 (2000), 203–211. https://doi.org/10.1016/S0024-3795(00)00048-3 doi: 10.1016/S0024-3795(00)00048-3
[8]	P. Chansangiam, Adjointations of operator inequalities and characterizations of operator monotonicity via operator means, Commun. Math. Appl., 7 (2016), 93–103. https://doi.org/10.26713/cma.v7i2.372 doi: 10.26713/cma.v7i2.372
[9]	S. Drury, Principal powers of matrices with positive definite real part, Linear Multilinear A., 63 (2015), 296–301. https://doi.org/10.1080/03081087.2013.865732 doi: 10.1080/03081087.2013.865732
[10]	S. Drury, M. Lin, Singular value inequalities for matrices with numerical ranges in a sector, Oper. Matrices, 8 (2014), 1143–1148. https://dx.doi.org/10.7153/oam-08-64 doi: 10.7153/oam-08-64
[11]	A. Ghazanfari, S. Malekinejad, Heron means and Pólya inequality for sector matrices, Bull. Math. Soc. Sci. Math. Roumanie Tome, 64 (2021), 329–339.
[12]	R. Horn, C. Johnson, Matrix analysis, Cambridge: Cambridge University Press, 2013.
[13]	H. Jafarmanesh, M. Khosravi, A. Sheikhhosseini, Some operator inequalities involving operator monotone functions, B. Sci. Math., 166 (2021), 102938. https://doi.org/10.1016/j.bulsci.2020.102938 doi: 10.1016/j.bulsci.2020.102938
[14]	M. Lin, Some inequalities for sector matrices, Oper. Matrices, 10 (2016), 915–921. https://dx.doi.org/10.7153/oam-10-51 doi: 10.7153/oam-10-51
[15]	M. Lin, Extension of a result of Hanynsworth and Hartfiel, Arch. Math., 104 (2015), 93–100. https://doi.org/10.1007/s00013-014-0717-2 doi: 10.1007/s00013-014-0717-2
[16]	M. Lin, F. Sun, A property of the geometric mean of accretive operators, Linear Multilinear A., 65 (2017), 433–437. https://doi.org/10.1080/03081087.2016.1188878 doi: 10.1080/03081087.2016.1188878
[17]	J. Liu, J. Mei, D. Zhang, Inequalities related to the geometric mean of accretive matrices, Oper. Matrices, 15 (2021), 581–587. https://doi.org/10.7153/oam-2021-15-39 doi: 10.7153/oam-2021-15-39
[18]	L. Nasiri, S. Fruichi, On a reverse of the Tan-Xie inequality for sector matrices and its applications, J. Math. Inequal., 15 (2021), 1425–1434. https://doi.org/10.7153/jmi-2021-15-97 doi: 10.7153/jmi-2021-15-97
[19]	M. Raissouli, M. Moslehian, S. Furuichi, Relative entropy and Tsallis entropy of two accretive operators, C. R. Math., 355 (2017), 687–693. https://doi.org/10.1016/j.crma.2017.05.005 doi: 10.1016/j.crma.2017.05.005
[20]	F. Tan, H. Chen, Inequalities for sector matrices and positive linear maps, Electron. J. Linear Al., 35 (2019), 418–423. https://doi.org/10.13001/ela.2019.5239 doi: 10.13001/ela.2019.5239
[21]	F. Tan, A. Xie, An extension of the AM-GM-HM inequality, Bull. Iran. Math. Soc., 46 (2020), 245–251. https://doi.org/10.1007/s41980-019-00253-z doi: 10.1007/s41980-019-00253-z
[22]	Y. Wang, J. Shao, Some logarithmic submajorisations and determinant inequalities for operators with numerical ranges in a sector, Ann. Funct. Anal., 12 (2021), 27. https://doi.org/10.1007/s43034-021-00117-w doi: 10.1007/s43034-021-00117-w
[23]	C. Yang, F. Lu, Inequalities for the Heinz mean of sector matrices involving positive linear maps, Ann. Funct. Anal., 11 (2020), 866–878. https://doi.org/10.1007/s43034-020-00070-0 doi: 10.1007/s43034-020-00070-0
[24]	F. Zhang, A matrix decomposition and its applications, Linear Multilinear A., 63 (2015), 2033–2042. https://doi.org/10.1080/03081087.2014.933219 doi: 10.1080/03081087.2014.933219

Reader Comments

Your name:*

Email:*
© 2022 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)