Operator inequalities via accretive transforms

Abstract

In this article, we employ certain properties of the transform $C_{M,m}(A)=(MI-A^*)(A-mI)$ to obtain new inequalities for the bounded linear operator $A$ on a complex Hilbert space $\mathcal{H}$. In particular, we obtain new relations among $|A|,|A^*|,|\mathfrak{R}A|$ and $|\mathfrak{I}A|$. Further numerical radius inequalities that extend some known inequalities will be presented too.

Keywords

• numerical radius
• norm inequality
• accretive operators

References

1. [1] Y. Bedrani, F. Kittaneh and M. Sababheh, From positive to accretive matrices, Positivity 25, 1601–1629, 2021.
2. [2] Y. Bedrani, F. Kittaneh and M. Sababheh, Numerical radii of accretive matrices, Linear Multilinear Algebra 69, 957–970, 2021.
3. [3] Y. Bedrani, F. Kittaneh and M. Sababheh, On the weighted geometric mean of accretive matrices, Ann. Funct. Anal. 12 (1), 2, 2021.
4. [4] Y. Bedrani, F. Kittaneh and M. Sababheh, Accretive matrices and matrix convex functions, Results Math. 77, 52, 2022.
5. [5] R. Bhatia, Matrix analysis, Springer-Verlag, New York, 1997.
6. [6] R. Bhatia, Positive definite matrices, Princeton Univ. Press, Princeton, 2007.
7. [7] P. Bhunia, S.S. Dragomir, M.S. Moslehian and K. Paul, Lectures on numerical radius inequalities, Infosys Science Foundation Series in Mathematical Sciences, Springer Cham, 2022.
8. [8] R. Bhatia and F. Kittaneh, Notes on matrix arithmetic-geometric mean inequalities, Linear Algebra Appl. 308, 203–211, 2000.

9. [9] M.D. Choi, A Schwarz inequality for positive linear maps on $C^*$-algebras, Illinois J. Math. 18, 565–574, 1974.
10. [10] C. Davis, A Schwartz inequality for convex operator functions, Proc. Amer. Math. Soc. 8, 42–44, 1957.
11. [11] S.S. Dragomir, New inequalities of the Kantorovich type for bounded linear operators in Hilbert spaces, Linear Algebra Appl. 428, 2750–2760, 2008.
12. [12] S.S. Dragomir, Inequalities for the numerical radius of linear operators in Hilbert spaces, Springer Briefs in Mathematics, Springer Cham, 2013.
13. [13] S. Drury, Principal powers of matrices with positive definite real part, Linear Multilinear Algebra 63, 296–301, 2015.
14. [14] S. Drury and M. Lin, Singular value inequalities for matrices with numerical ranges in a sector, Oper. Matrices 8, 1143–1148, 2014.
15. [15] C.-K. Fong and J.A.R. Holbrook, Unitarily invariant operator norms, Can. J. Math. 35, 274–299, 1983.
16. [16] T. Furuta, J. Micic Hot, J. Pecaric and Y. Seo, Mond-Pecaric method in operator inequalities, Monographs in Inequalities 1, Element, Zagreb, 2005.
17. [17] K.E. Gustafson and D.K.M. Rao, Numerical range, Springer, New York, 1997.
18. [18] P.R. Halmos, A Hilbert space problem book, 2nd ed., Springer, New York, 1982.
19. [19] R.A. Horn and C.R. Johnson, Topics in matrix analysis, Cambridge University Press, 1991.
20. [20] F. Kittaneh, Numerical radius inequalities for Hilbert space operators, Studia Math. 168, 73–80, 2005.
21. [21] M. Lin, Some inequalities for sector matrices. Oper. Matrices 10, 915–921, 2016.
22. [22] H.R. Moradi, S. Furuichi and M. Sababheh, Some operator inequalities via convexity, Linear Multilinear Algebra 70 (22), 7740–7752, 2022. Doi: 10.1080/03081087.2021.2006592.
23. [23] H.R. Moradi, M.E. Omidvar, I.H. Gümüs and R. Naseri, A note on some inequalities for positive linear maps, Linear Multilinear Algebra 66 (7), 1449–1460, 2018.
24. [24] M. Niezgoda, Accretive operators and Cassels inequality, Linear Algebra Appl. 433, 136–142, 2010.
25. [25] M. Raïssouli, M.S. Moslehian and S. Furuichi, Relative entropy and Tsallis entropy of two accretive operators, C. R. Acad. Sci. Paris Ser. I 355, 687–693, 2017.
26. [26] P.Y. Wu and H.-L. Gau, Numerical ranges of Hilbert space operators, Encyclopedia of Mathematics and its Applications, 179, Cambridge University Press, Cambridge, 2021.