Abstract
In this paper, we obtain some upper bounds for the singular values of sums of product of matrices. The obtained forms involve direct sums and mean-like matrix quantities. As applications, several bounds will be found in terms of the Aluthge transform, matrix means, matrix monotone functions and accretive-dissipative matrices. For example, we show that if X is an \(n\times n\) accretive-dissipative matrix, then
for \(j=1,2,\ldots n\), where \(s_j(\cdot ), \Re (\cdot )\) and \(\Im (\cdot )\) denote the \(j-\)th singular value, the real part and the imaginary part, respectively. We also show that if \(\sigma _f,\sigma _g\) are two matrix means corresponding to the operator monotone functions f, g, then
for \(j =1,2, \ldots , n\), where A, B are two positive definite \(n\times n\) matrices.
Similar content being viewed by others
Availability of data and materials
Not applicable.
References
Albadawi, H.: Singular value and arithmetic-geometric mean inequalities for operators. Ann. Funct. Anal. 3, 10–18 (2012)
Aluthge, A.: On \(p\)-hyponormal operators for \(0<p<1\). Integral Equ. Oper. Theory 13, 307–315 (1990)
Ando, T.: Topics on operator inequalities. Lecture Note, Sapporo (1978)
Audeh, W., Kittaneh, F.: Singular value inequalities for compact operators. Linear Algebra Appl. 437, 2516–2522 (2012)
Bhatia, R., Davis, C.: More matrix forms of the arithmetic-geometric mean inequality. Siam J. Matrix Anal. Appl. 14, 132–136 (1993)
Bhatia, R., Kittaneh, F.: On the singular values of a product of operators. SIAM J. Matrix Anal. Appl. 11, 272–277 (1990)
Bhatia, R., Kittaneh, F.: The matrix arithmetic-geometric mean inequality revisited. Linear Algebra Appl. 428, 2177–2191 (2008)
Cho, M., Tanahashi, K.: Spectral relations for Aluthge transform. Sci. Math. Jpn. 55, 77–83 (2002)
Hirzallah, O., Kittaneh, F.: Inequalities for sums and direct sums of Hilbert space operators. Linear Algebra Appl. 424, 71–82 (2007)
Hou, J.C., Du, H.K.: Norm inequalities of positive operator matrices. Integral Equ. Oper. Theory 22, 281–294 (1995)
Kittaneh, F.: Norm inequalities for sums of positive operators. J. Operator Theory 48(1), 95–103 (2002)
Kittaneh, F.: Norm inequalities for sums and differences of positive operators. Linear Algebra Appl. 383, 85–91 (2004)
Kittaneh, F.: Norm inequalities for sums of positive operators. II, Positivity 10, 251–260 (2006)
Kubo, F., Ando, T.: Means of positive linear operators. Math. Ann. 246, 205–224 (1980)
Tao, Y.: More results on singular value inequalities of matrices. Linear Algebra Appl. 416, 724–729 (2006)
Zhan, X.: Singular values of difference of positive semi-definite matrices. SIAM J. Matrix Anal. Appl. 22, 819–823 (2000)
Zhao, J.: Singular value and unitarily invariant norm inequalities for sums and products of operators, Adv. Oper. Theory 6, Article ID. 64 (2021)
Zou, L.: An arithmetic-geometric mean inequality for singular values and its applications. Linear Algebra Appl. 528, 25–32 (2017)
Acknowledgements
Not applicable.
Funding
Not applicable.
Author information
Authors and Affiliations
Contributions
Authors declare that they have contributed equally to this paper. All authors have read and approved this version.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Kittaneh, F., Moradi, H.R. & Sababheh, M. Singular value inequalities with applications to norms and means of matrices. Acta Sci. Math. (Szeged) (2024). https://doi.org/10.1007/s44146-024-00113-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s44146-024-00113-1
Keywords
- Singular value
- Direct sum
- Unitarily invariant norm
- Matrix mean
- Aluthge transform
- Accretive-dissipative matrix