Adjointations of Operator Inequalities and Characterizations of Operator Monotonicity via Operator Means

Authors

  • Pattrawut Chansangiam Department of Mathematics, Faculty of Science, King Mongkut's Institute of Technology Ladkrabang, Chalongkrung Rd., Bangkok 10520

DOI:

https://doi.org/10.26713/cma.v7i2.372

Keywords:

Operator mean, Operator monotone function, Operator inequality

Abstract

We propose adjointations between operator orderings, which convert any operator inequalities/identities associated with certain binary operations to new ones. Then we prove that a continuous function \(f:(0,\infty) \to (0,\infty)\) is operator monotone increasing if and only if \(f(A \: !_t \: B) \leq f(A) \: !_t \: f(B)\) for any positive operators \(A,B\) and scalar \(t \in [0,1]\). Here, \(!_t\) denotes the \(t\)-weighted harmonic mean. As a counterpart, \(f\) is operator monotone decreasing if and only if the reverse of preceding inequality holds. Moreover, we obtain many characterizations of operator monotone increasingness/decreasingness in terms of operator means. These characterizations lead to many operator inequalities involving means.

Downloads

Download data is not yet available.

References

T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Alg. Appl. 26 (1979), 203–241.

T. Ando and F. Hiai, Operator log-convex functions and operator means, Math. Ann. 350 (2011), 611–630.

J.S. Aujla, M.S. Rawla and H.L. Vasudeva, Log-convex matrix functions, Ser. Mat. 11 (2000), 19–32.

R. Bhatia, Positive Definite Matrices, Princeton University Press, New Jersey (2007).

P. Chansangiam, Positivity, betweenness, and strictness of operator means, Abstr. Appl. Anal. Article ID 851568, (2015), 5 pages, doi:10.1155/2015/851568.

F. Hansen and G.K. Pedersen, Jensen's inequality for operators and Löwner's theorem, Math. Ann. 258 (1982), 229–241.

F. Hiai, Matrix analysis: matrix monotone functions, matrix means, and majorizations, Interdiscip. Inform. Sci. 16 (2010), 139–248.

F. Hiai and D. Petz, Introduction to Matrix Analysis and Applications, Springer, New Delhi (2014).

S. Izumino and N. Nakamura, Operator monotone functions induced from Löwner-Heinz inequality and strictly chaotic order, J. Math. Ineq. 7 (2004), 103–112.

F. Kubo and T. Ando, Means of positive linear operators, Math. Ann. 246 (1980), 205–224.

K. Löwner, íœber monotone matrix funktionen, Math. Z. 38 (1934), 177–216.

X. Zhan, Matrix Inequalities, Springer-Verlag, New York (2002).

Downloads

Published

04-11-2016
CITATION

How to Cite

Chansangiam, P. (2016). Adjointations of Operator Inequalities and Characterizations of Operator Monotonicity via Operator Means. Communications in Mathematics and Applications, 7(2), 93–103. https://doi.org/10.26713/cma.v7i2.372

Issue

Vol. 7 No. 2 (2016)

Section

Research Article

License

Creative Commons Licence 4.0 by  

Authors who publish with this journal agree to the following terms:

  • Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  • Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  • Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.