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

Pattrawut Chansangiam

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.

Keywords


Operator mean; Operator monotone function; Operator inequality

Full Text:

PDF

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, Über monotone matrix funktionen, Math. Z. 38 (1934), 177–216.

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


Refbacks

  • There are currently no refbacks.


eISSN 0975-8607; pISSN 0976-5905