zbMATH — the first resource for mathematics

Eigenvalue inequalities for differences of means of Hilbert space operators. (English) Zbl 1248.47019
Given bounded linear operators \(A,B\) on a Hilbert space such that \(A\geq B>O \) and \(A-B\) is compact, and a scalar \(\mu\in (0,1)\), the authors consider the \(\mu\)-weighted arithmetic (resp., geometric) mean \(A\nabla _{\mu} B = (1-\mu)A+\mu B\) (resp., \(A\sharp _{\mu}B=A^{1/2}(A^{-1/2}BA^{-1/2})^{\mu}A^{1/2}\)) and the \(\mu\)-weighted Heinz mean \(H_{\mu}(A,B)=1/2(A\sharp _{\mu}B +A\sharp _{1-\mu}B)\). The estimates for the singular values of the differences \(A\nabla _{\mu} B -A\sharp _{\mu}B\), \((A\nabla _{\mu} B)A^{-1}(A \nabla _{\mu} B) -A\sharp _{2 \mu}B \), \(A\nabla _{\mu}(BA^{-1}B)-A\sharp _{2 \mu}B \), \(B\nabla _{\mu}(AB^{-1}A)-B\sharp _{2 \mu}A\) and \(\frac{A+B}{2} - H_{\mu}(A,B)\) are expressed in terms of the singular values of the operators \(A^{-1/2}(A-B)^{2}A^{-1/2}\) or \(B^{-1/2}(A-B)^{2}B^{-1/2}\). Equality conditions for the corresponding inequalities are also obtained.

47A64 Operator means involving linear operators, shorted linear operators, etc.
Full Text: DOI
[1] Ando, T., Majorizations and inequalities in matrix theory, Linear algebra appl., 199, 17-67, (1994) · Zbl 0798.15024
[2] Bhatia, R., Matrix analysis, GTM169, (1997), Springer Verlag New York
[3] Furuta, T.; Mićić Hot, J.; Pečarić, J., Mond – pečarić method in operator inequalities, (2005), Element Zagreb · Zbl 1135.47012
[4] Gohberg, I.C.; Krein, M.G., Introduction to the theory of linear nonself-adjoint operators, Transl. math. monogr., vol. 18, (1969), AMS, Providence RI · Zbl 0181.13504
[5] I.H. Gumus, O. Hirzallah, N. Taskara, Singular value inequalities for the arithmetic, geometric and Heinz means of matrices, Linear and Multilinear Algebra, in press. · Zbl 1242.15015
[6] Hirzallah, O.; Kittaneh, F., Norm inequalities for weighted power means of operators, Linear algebra appl., 341, 181-193, (2002) · Zbl 1017.47003
[7] O. Hirzallah, F. Kittaneh, M. Krnić, N. Lovričević, J. Pečarić, On some strengthened refinements and converses of means inequalities for Hilbert space operators, preprint.
[8] F. Kittaneh, M. Krnić, N. Lovričević, J. Pečarić, Improved arithmetic – geometric and Heinz means inequalities for Hilbert space operators, Publ. Math. Debrecen, in press.
[9] Kittaneh, F.; Manasrah, Y., Reverse Young and Heinz inequalities for matrices, Linear and multilinear algebra, 59, 1031-1037, (2011) · Zbl 1225.15022
[10] Kubo, F.; Ando, T., Means of positive linear operators, Math. ann., 246, 205-224, (1980) · Zbl 0412.47013
[11] Merris, R.; Pierce, S., Monotonicity of positive semidefinite Hermitian matrices, Proc. amer. math. soc., 31, 437-440, (1972) · Zbl 0209.06404
[12] Mitrinović, D.S.; Pečarić, J.; Fink, A.M., Classical and new inequalities in analysis, (1993), Kluwer Acad. Publ. Dordrecht, Boston, London · Zbl 0771.26009
[13] Ringrose, J.R., Compact non-self-adjoint operators, (1971), Van Nostrand Reinhold New York · Zbl 0223.47012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.